介绍
Introduction
数学向我们讲述了哪种现实？人们普遍认为，数学家全神贯注于抽象公式，只是出于莫名其妙的原因，这些公式才适用于科学的各个领域。
About which reality does mathematics speak to us? It is widely supposed that mathematicians preoccupy themselves with abstract formulas, and that it is only for inexplicable reasons that these formulas have applications in every area of science.
我们设想的非物质实体似乎随后注定要定义世界上实际发生的现象模型。一方面，有真实的、现在的事物；另一方面，数学概念，我们头脑的创造，以或多或少有效的方式模拟他们的行为。对公式和方程的描述能力背后的真正原因的无知当然无助于阐明数学思维背后的潜在动机。相反，它为数学家不倾向于与世界接触的想法提供了货币。数学继续将自己呈现为一门科学，它用规则和概念来阐述巧妙的操作，这些规则和概念似乎是为了正确执行而被召唤出来的。1对自然现象的观察提出了其中一些想法，这无关紧要。这些操作迅速产生了先进而复杂的概念，这些概念与可观察的现实相距甚远，并最终证实了数学的扭曲形象只是一种语言游戏或一系列空洞的公式。
We conceive of immaterial entities that seem subsequently to be destined to define models of phenomena that actually occur in the world. On the one hand, there are real, present things; on the other, mathematical concepts, creations of our mind which simulate their behaviour in a more or less effective way. Ignorance of the true reason behind the descriptive power of formulas and equations certainly doesn’t help to clarify the underlying motivation behind mathematical thinking. It gives currency instead to the idea that mathematicians are not inclined to engage with the world. Mathematics continues to present itself as a science that elaborates ingenious operations with rules and concepts that seem to have been conjured up with the sole aim of their being executed correctly.1 It matters little that some of these ideas have been suggested by the observation of natural phenomena; the operations rapidly produce advanced and complex concepts that distance themselves from observable reality and ultimately confirm the distorted image of mathematics as a merely linguistic game or a series of empty formulas.
然而，如果我们转向它的遥远历史和它最深刻的目的，数学的方向似乎与通常假设的截然不同。它的起源允许我们要明白，古代算术和几何学开始承担的角色与其说是描述或模拟真实事物，不如说是为它们所参与的现实提供基础。是具体的事物本身——那些可以直接和立即感知的事物——是变化的和可变的，因此容易显得不真实。为了准确地找到使它们摆脱这种不稳定和消逝的确切原因，人们不得不转而关注数字、它们的关系和几何图形。
If we turn to its remote history and to its deepest purpose, however, mathematics appears to be orientated very differently than is commonly supposed. Its origins allow us to understand that ancient arithmetic and geometry were beginning to assume the role not so much of describing or simulating real things as offering a foundation for the very reality of which they were a part. It was concrete things themselves – those that were directly and immediately perceivable – that were shifting and mutable, and therefore prone to appear unreal. To find precisely what removed them from such instability and evanescence, one had to look instead to numbers, to their relations and to the figures of geometry.
如果我们想一想芝诺著名的悖论、毕达哥拉斯学派和古代原子论者的数点、柏拉图的数学哲学、不可通约性的发现以及关系性概念的意义（lógos)，关于巴比伦微积分和吠陀数学，我们面临着大量的知识，旨在捕捉自然界中存在的事物最内在、最不可见——以及最真实——的方面。19 世纪详细阐述的数论和连续数学理论将其自身呈现为古代毕达哥拉斯主义的理想延续，以及由对现实的原子性质的直觉所激发的世界观。当时的数学家继续争辩说，他们的符号结构对应于非常真实的实体，而普遍的印象是，他们的理论的成功取决于理解世界所必需的知识基础。
If we think about Zeno’s famous paradoxes, the number-points of the Pythagoreans and the atomists of antiquity, about Plato’s mathematical philosophy, the discovery of incommensurability and the significance of the concept of relationality (lógos), about Babylonian calculus and Vedic mathematics, we are faced with a great mass of knowledge designed to capture the most internal and invisible – as well as the most real – aspect of the things that exist in nature. The theory of numbers and of continuous mathematics elaborated in the nineteenth century presented itself as the ideal continuation of ancient Pythagoreanism, and of a vision of the world inspired by an intuition of the atomistic nature of reality. Mathematicians of the period, then, continued to contend that their symbolic constructions corresponded to very real entities, and the widespread impression was that on the success of their theories depended the foundation of knowledge that was necessary for understanding the world. When in the early twentieth century the principles of these theories became uncertain and began to undergo a critical revision, mathematics was obliged to search for the reasons that make a system of calculation genuinely concrete and reliable.
这时一个关键术语开始在数学家中持续流传：算法，这个词表示并非如此作为一个实际过程的抽象公式。2根据机器预测的模式，这个过程需要在有限数量的步骤中展开，从初始数据集到空间和时间的最终结果。基于递归、图灵机或其他公式的“算法”的正式定义可以追溯到上个世纪的第四个十年——但第一个迹象表明，算法的概念将继承数学的意义现实——也就是说，所有数学家认为是真实的和现实的——在 20 世纪头十年，在数学直觉主义的早期迹象和法国数学家埃米尔·博雷尔所面临的第一个论点中已经被见证了语义悖论和基本原理的初期危机。
At this time a key term began to circulate persistently among mathematicians: algorithm, a word which denoted not so much an abstract formula as an actual process.2 This process needed to unfold in a finite number of steps, from an initial set of data to a final outcome in space and time, according to the modalities predicted by a machine. The formal definition of ‘algorithm’, based on recursion, on Turing’s machine or on other formulations, dates back to the fourth decade of the last century – but the first indications that it would be this concept of algorithm that would inherit the sense of mathematical reality – that is to say, all that mathematicians consider to be real and actual – were already being witnessed in the first decade of the twentieth century, in the early signs of mathematical intuitionism and in the first arguments with which the French mathematician Émile Borel confronted the semantic paradoxes and incipient crisis of fundamentals.
算法科学沿着跨越整个世纪的动荡发展弧线，在 30 年代的正式定义中达到顶峰，然后在构建第一台数字计算器之后分成两个不同但互补的趋势：一方面，信息理论，具有可计算性和计算复杂性的抽象概念；另一方面，一门大规模的微积分科学，致力于以纯算术和数值的形式解决物理、经济学、工程和计算机科学中的数学问题。这第二个趋势的多个哲学方面尚未得到充分分析，但它在生活的各个领域、文化和社会组织中的贡献已经是显而易见的，
The science of algorithms followed a tumultuous arc of development that spanned the entire century, reaching a culmination in the formal definitions of the thirties, before bifurcating into two different but complementary trends after the construction of the first digital calculators: on the one hand, information theory, with its abstract notions of computability and of computational complexity; on the other, a science of calculus on a large scale, dedicated to resolving mathematical problems in physics, economics, engineering and computer science in purely arithmetical and numerical terms. The multiple philosophical facets of this second trend have not yet been sufficiently analysed, but it is already evident to all how much it has contributed in every area of life, to culture and social organization, with the multiplication of a diverse variety of calculation processes aimed at solving specific problems of the most varied kind.
在大规模数值计算中，算法的理论有效性旨在实现计算效率。今天似乎很清楚，为了成为真实的，通过计算过程构建的完全相同的数学实体必须能够以与有效算法相同的方式被考虑。今天，效率首先取决于它们促进计算复杂性和计算错误增长的方式。特别是，误差取决于计算过程中数字增长的速度。
In numerical calculation on a large scale the theoretical effectiveness of algorithms aims at achieving computational efficiency. And today it seems clear that, in order to be real, the very same mathematical entities that have been constructed through a process of calculation must be capable of being thought of in the same way as efficient algorithms. Today the efficiency depends above all on the way in which they facilitate growth in computational complexity and errors of calculation. In particular, the error depends on how rapidly the numbers grow in the course of calculation.
数字增长的原因是严格的数学计算，并且可以通过相对先进的定理进行分析。但值得注意的是，这种增长的原因，在其所有方面，已经是古代思想中最仔细研究的对象，而这正是希腊几何学中处理数量增长的方式，在吠陀计算和美索不达米亚算术有助于理解现代算法中数字增长的原因。其原因既简单又令人惊讶：从那时起，一些重要的计算模式一直保持不变，直到今天大规模计算所采用的最复杂的策略。
The reasons for the growth in the numbers are strictly mathematical and may be analysed thanks to relatively advanced theorems. But it is worth noting that the reason for this growth, in all of its aspects, was already the object of the closest scrutiny in ancient thought, and it is precisely the way in which the growth of quantities is treated in Greek geometry, in Vedic calculations and in Mesopotamian arithmetic that has contributed to the understanding of the causes of the growth of numbers in modern algorithms. The reason for this is as simple as it is surprising: some important computational schemas have remained unchanged since then, right up until the most complex strategies of which large-scale calculation avails itself today.
这些模式从何而来？在某些与现代科学特别相关的情况下，来源很清楚：这些图式源自人类设计和神圣命令的单一组合。在吠陀印度，火神阿格尼的祭坛具有复杂的几何形状，需要能够通过使用在希腊几何和美索不达米亚计算中也可以找到的特定技术来扩大一百倍而不改变形状。在希腊，就像将立方体加倍的情况一样，神也要求扩大形状。但是几何形式的扩大与委派的算法密切相关，以近似那些当面对必须测量几何大小（例如正方形的对角线或周长与直径之间的关系）时，这些数字是不合理的。远在笛卡尔之神之前，是吠陀诸神和希腊诸神，他们保证了神秘主义与自然之间、我们最亲密的领域与外部现实之间的联系。在那个阶段，数学也提供了这种可能联系的原理。无论如何，受宗教信仰启发的古代几何学的增长方式今天反映在数字计算中的数字增长中，对计算本身的稳定性和数学模型的预测能力产生了重要影响。事实上，几何的增长方式数字，特别是正方形，通常与生成分数p / q的数值程序相关，这些分数近似于无理数，其中p和q是整数。但通常p和q增长得越快，方法收敛越快，对整个计算过程的精度和稳定性都有潜在的负面影响。
Where do these schemas derive from? In certain cases of particular relevance to modern science, the sources are clear: these schemas derive from a singular combination of human design and divine dictate. In Vedic India the altars of Agni, the fire god, had complex geometrical forms and needed to be capable of being enlarged a hundredfold without changing shape, by using specific techniques that can also be found in Greek geometry and Mesopotamian calculation. In Greece it so happened, as in the case of doubling the cube, that the enlargement of a form was also demanded by a deity. But the enlargement of the geometric form was in strict relation to the algorithms delegated to approximate those numbers which, when faced with having to measure geometric magnitudes such as the diagonal of a square, or the relation between a circumference and diameter, are irrational. It was the Vedic gods and those of Greece, long before the God of Descartes, who guaranteed a nexus between mysticism and nature, between our most intimate sphere and external reality. Mathematics offered at that stage, too, the principle of this possible connection. At any rate, the modalities of growth in ancient geometry, inspired by religious observance, are reflected today in the growth of numbers in digital calculations, having an essential impact on the stability of calculation itself and on the predictive power of mathematical models. In fact, the modalities of growth of geometric figures, in particular the square, are often correlated to numerical procedures that generate fractions p/q, which approximate irrational numbers, where p and q are whole numbers. But usually p and q grow much more rapidly the more rapid the convergence in method, with potential negative effects on the precision and stability of the entire process of calculation.
无理数是实实在在的实体，其本体论地位可与整数相媲美，这是 19 世纪末数学的一项成就，也是算术连续统概念在时间。但是，算法和数字计算科学的发展在 20 世纪成为一种新的对立的表达：一种常年紧张的最终行为——在芝诺的悖论中已经涉及——数字和几何之间，离散之间和连续的。谈论它们之间的冲突是合理的，因为从 20 世纪初开始，算法的研究就受到了思想较量的推动。旨在重新评估数学中最现实和最有建设性的方面，与那些引起悖论和基本面危机的抽象形成鲜明对比：一方面，Émile Borel 的精湛指挥标志着定义数学实体的重要性通过算法构造；另一方面，由 L. E. J. Brouwer 和数学纲要中的数学直觉主义表达的戏剧性分裂。Brouwer 认为一个数字只有在建立时才存在，他对盛行的科学体系提出了普遍挑战，质疑经典分析的基本定义。
The thesis that irrational numbers are real entities, with an ontological status comparable to that of whole numbers, was an achievement of the mathematics of the end of the nineteenth century, and of the way in which the concept of the arithmetical continuum was defined at the time. But the development of the science of algorithms and digital calculation became in the twentieth century the expression of a new kind of opposition: a kind of final act of the perennial tension – already touched on in Zeno’s paradoxes – between numbers and geometry, between the discrete and the continuous. It is legitimate to speak of a clash between them because the study of algorithms was propitiated from the very beginning of the twentieth century by a contest of ideas aimed at re-evaluating the most realistic and constructive aspects of mathematics, in contradistinction to those abstractions which had given rise to paradoxes and a crisis of fundamentals: on the one hand, the masterly command of Émile Borel which signalled the importance of defining mathematical entities through algorithmic constructions; on the other, the dramatic schism articulated by L. E. J. Brouwer and by mathematical intuitionism within the compendium of mathematics. Arguing that a number exists only if it is built, Brouwer issued a general challenge to the prevailing scientific system, calling into question the fundamental definitions of classical analysis.
建构主义哲学以实际可计算性为基础，赋予了似乎与数学的抽象使命格格不入的事物新的卓越地位。他们重视具体操作、自然的真实性，以及最终在空间和时间允许的范围内运行的机器内部发展的计算过程。但同样明显的是，重要的计算策略是根据人类在那些密切参与假设与神灵交流的时代所制定的相同模式建模的。出于仪式的目的，在吠陀印度和希腊，规模增长的原因是至关重要的，并且必须在数学上面对。扩大几何形状的基本蓝图仍然可以在最先进的计算数学中追溯。这些图式并没有改变，尽管它们肯定已经通过复杂的数学理论进行了阐述和完善。从这些理论中，我们还得出了它们的效率和将自然数学模型转化为纯数字信息的有效能力的基本原理。
The constructivist philosophies, based on the idea of actual calculability, have assigned new pre-eminence to that which seemed alien to the abstract vocation of mathematics; they have given importance, that is, to the concrete operation, the factuality of nature and, ultimately, to the computational process that develops inside a machine operating within the limits allowed by space and time. But it is equally evident that important computational strategies are modelled on the same schemas that mankind had elaborated during those eras when they were closely engaged in supposed communications with the gods. For ritual purposes, in Vedic India as in Greece, the reason for growth in magnitude was of fundamental importance and had to be confronted mathematically. And the basic blueprints for enlarging a geometric shape are still traceable in the most advanced computational mathematics. The schemas have not changed, though they have certainly been elaborated and perfected through complex mathematical theories. From these theories we also derive the rationale for their efficiency and their effective capacity to translate mathematical models of nature into pure digital information.
相同的计算过程，在无数具体的自动操作中表达的相同过程，只能通过抽象的数学结构才能发生，这些结构或多或少是人为地插入到计算中的。数学抽象以必要和系统的方式与自动执行操作的重要性相结合。由于复杂的理论假设以及数字、函数和矩阵的特殊性质，使得计算成为可能。
The same computational process, the same process articulated in a myriad of concrete automatic operations, can take place only thanks to abstract mathematical structures, inserted more or less artificially into the calculation. Mathematical abstraction is combined in a necessary and systematic way with the materiality of the automatic execution of operations. The calculation is made possible because of complex theoretical presuppositions and the special properties of numbers, functions and matrices.
因此，问题仍然悬而未决：数字是真实的实体吗？如果我们的回答是肯定的，它们是否都以完全相同的方式“真实”？这两个问题需要一起解决。上个世纪的历史以及对数字和算法概念的分析让我们看到了一个初步结论：存在不同种类的数字，它们不具有相同的本体论地位，但与之相关的是可以归因于现实中的存在，出于各种原因和从不同的角度。确定数字真实性的一个关键标准是它们在计算过程中的增长方式。这种现象的第一个原因应该在古代思想中详细阐述的几何增长分析中寻找，特别是在希腊、吠陀和美索不达米亚的数学中。
The question thus remains open: are numbers real entities? And if we were to answer in the affirmative, are they all ‘real’ in exactly the same way? The two questions need to be tackled together. The history of the last century and an analysis of the concepts of number and algorithm allow us to see our way towards an initial conclusion: different kinds of numbers exist that do not have the same ontological status, but in relation to which it is possible to ascribe an existence in reality, for a variety of reasons and from different points of view. A key criterion for establishing the reality of numbers is the way in which they grow in the processes of calculation. And the first reason for this phenomenon should be sought in the analysis of geometrical growth elaborated in ancient thought, especially in Greek, Vedic and Mesopotamian mathematics.
1. 抽象、存在与现实
1. Abstraction, Existence and Reality
数学从何而来，它的目标是什么？为什么有三角形、正方形、圆形和五边形？什么样的现实和存在可以归因于数字？正如一些最顽固的形式主义者经常承认的那样，数学是真正的知识——我们可以自信地说，这种知识的对象不是任意的，它不依赖于反复无常的想象或某些公理的任意选择或原则。此外，它经常碰巧被视为一个外部现实，独立于阐述它的头脑。
Where does mathematics come from, and what are its objectives? Why are there triangles, squares, circles and pentagons? What kind of reality and existence may be attributed to numbers? Mathematics, as even some of the most intransigent formalists often admit, is real knowledge – and the object of this knowledge, we can say with confidence, is not arbitrary, does not depend on a capricious imagination or on the arbitrary choice of certain axioms or principles. Furthermore, it frequently happens to be perceived as an external reality, independent of the mind that elaborates it.
我们通常认为数学是一门抽象科学，因为在实践中，它从特定实体（例如数字）中提取关系和模式，以研究共同的属性——就好像这些属性反过来又是遵守其自身规律的新实体，其优点是关于后者可以说的所有内容都可以应用于从中进行抽象的不同特定实体。推理，当它是抽象的，因此变得更普遍和更有力。但是抽象使本质的识别变得更加困难数学实体的位置，易于定义的内在特征的位置。从它们的存在中，至少作为我们思想的对象，似乎无法得出某种稳定且清晰可辨的事物的真正本质。essentia和existentia之间传统的相互联系 ——如果另一个术语不存在，一个术语将无法获得真实的地位——似乎已经丢失的。1数字的本质是存在于某种我们可以直接理解的与它们有关的特殊性质，还是源自一个抽象领域的属性，在这个领域中，所讨论的数字只是一个可能的——不一定是唯一的——例子？
We usually think that mathematics is an abstract science, since in practice it extracts relationships and patterns from specific entities such as numbers in order to study common properties – as if the properties were in turn new entities that obey their own laws, with the advantage that all that may be said about the latter may be applied to the different specific entities from which the abstraction has been made. The reasoning, when it is abstract, thus becomes more general and more powerful. But abstraction makes more problematic the identification of an essence of mathematical entities, the location of an intrinsic character susceptible to definition. From their existence, at least as objects of our thought, a real essence of something that is stable and clearly recognizable does not seem to derive. The traditional reciprocal connection between essentia and existentia – whereby one term would not achieve the status of the real if the other were absent – seems to have been lost.1 Does the essence of numbers consist in some kind of special nature that pertains to them which we can apprehend directly, or does it derive rather from the properties of an abstract domain of which the numbers in question are only a possible – not necessarily unique – example?
通常是第二个受青睐的想法。单独研究的特性形成了一个从公理衍生的真理系统——数学，与任何形式的直接直觉相分离，将被视为一门纯粹形式关系的科学，独立于对它们的每一种具体解释。一个特定的数值域，2例如实数或复数的域，满足其他数学领域也符合的相同公理。因此，可能会发生一个数字域只能通过同构来识别，因为它与具有相同属性的其他数学实体无法区分。这种情况足以使数学完全不受对这些数字特定性质的可能搜索的影响。
It is usually the second idea that is favoured. The properties that are studied separately form a system of truths derived from axioms – and mathematics, dissociated from any form of direct intuition, would then be considered as a science of pure formal relations, independent of every concrete interpretation of them. A specific numerical field,2 such as that of real or complex numbers, satisfies the same axioms to which other mathematical fields also conform. It can thus happen that a number field may be identified only by an isomorphism, because it is indistinguishable from other mathematical entities that have the same properties. It’s a circumstance that is enough to make mathematics completely impervious to the possible search for the specific nature of those numbers.
数学对象的识别或表征通常取决于其与属性相关的规范，这些属性与任何可能的构造或表示无关。有时，基于几个属性的单个简单定义就足以识别整个类同构域。3
The recognition, or characterization, of a mathematical object usually depends on its specification in relation to properties that are independent of any possible construction or representation. And it is sometimes sufficient for a single, simple definition based on a few properties to identify an entire class of isomorphic domains.3
逻辑工具是否适合于确定一个数学实体是否以及在什么情况下以某种形式在独立于我们的外部世界中真实存在？在这个问题上存在着相互矛盾的观点。例如，戈特洛布·弗雷格（Gottlob Frege）和伯特兰·罗素（Bertrand Russell）在这个主题上的观点存在显着差异，引用逻辑主义的两位杰出代表，这激发了试图证明所有数学都可以还原为逻辑。对于弗雷格来说，数字是必须以某种方式定义的逻辑对象。它们不是通过定义创造出来的：定义仅仅展示了自身存在的东西。4罗素的立场不同——显然更倾向于唯名论，但并非严格如此。对于罗素来说，逻辑作为一个整体是了解外部世界不可或缺的工具。在《我们对外部世界的知识》（1914）中，他写道，解释事件本质的基本要素是事物、品质和关系，以前被逻辑忽略，它假设所有陈述都应该有一个主谓形式。传统逻辑没有考虑到现实关系，实际上对于描述世界是必不可少的，并且是消除传统形而上学的错误所必需的。根据罗素的说法，很可能正是传统的神秘主义和形而上学提出了我们所感知的世界不真实的想法。罗素继续解释说，逻辑用于阐明对世界的描述，这种描述起源于原子命题，这些命题记录了经验经验的事实。从原子命题我们转向更复杂的命题，这总是要归功于逻辑。逻辑显然不进入基本事实的登记，而是构成一种全面的理解，先天的在性质上，所有潜在的扣除都基于此。这种演绎的复合体，虽然不是完全来自感官知识，但应该被认为是真实有效的知识。罗素总结说，数学是其中的重要组成部分。
Are the instruments of logic suitable for establishing whether, and under what circumstances, a mathematical entity has real existence in some form or other, in an external world that is independent of us? There are conflicting views on this matter. There is, for instance, a significant difference between the views of Gottlob Frege and Bertrand Russell on this subject, to cite two of the pre-eminent exponents of logicism, which inspired the idea which sought to demonstrate that all mathematics is reducible to logic. For Frege, numbers were logical objects that one must define in some way. They are not created through definition: the definition merely demonstrates what exists in its own right.4 Russell’s position is different – decidedly more inclined towards nominalism, but not rigidly so. For Russell, logic as a whole is an indispensable instrument of knowledge of the external world. In Our Knowledge of the External World (1914), he wrote that the fundamental elements for explaining the nature of events are things, qualities and relationships, previously ignored by logic, which had assumed that all statements should have a subject–predicate form. Traditional logic had not taken into account the reality of relations, which were in fact indispensable to descriptions of the world and were needed in order to dissipate the errors of traditional metaphysics. In all likelihood, according to Russell, it was precisely traditional mysticism and metaphysics which had posited the idea of the unreality of the world that we perceive. Logic, Russell goes on to explain, serves to articulate a description of the world that originates with atomic propositions, which register the facts of empirical experience. From the atomic propositions we move to more complex ones, always thanks to logic. Logic obviously does not enter into the registration of elementary facts, but constitutes a comprehensive understanding, a priori in character, on which all potential deductions are based. This complex of deductions, while not deriving exclusively from sensory knowledge, should be considered to be knowledge that is real and effective. And mathematics, Russell concludes, is an essential part of it.
然而，如果我们注意逻辑实际上告诉我们什么，在各个阶段似乎占主导地位的是最激进的唯名论。拉塞尔无法避免承认定义数字的范畴不是我们必须确定其存在的事物。当然，使用逻辑语言，人们努力声明一个数字、一个集合或一个函数存在。但是数字的真实性与这种语言所确立的存在完全不相符。断言一个数学实体存在的逻辑命题“不是为了成功地了解存在的东西，而是为了知道属于我们或他人的给定断言或给定论题所说的是什么；这是正确的语言问题，而不是本体论问题”。5
Nevertheless, if we pay attention to what logic actually tells us, at stages what seems to prevail is the most radical nominalism. Russell could not avoid acknowledging that the categories which define numbers are not things of which we have to establish the existence. Of course, with the language of logic, one strives to declare that a number, a set or a function exists. But the reality of numbers does not coincide at all with the existence established by this language. The logical propositions which assert that a mathematical entity exists serve ‘not so as to succeed in knowing that which exists, but so as to know what a given assertion or given thesis, belonging to us or to others, says that there is; and this is properly speaking a linguistic problem, not an ontological one’.5
仅靠逻辑不足以建立抽象对象的本体。我们也不应该对古德曼和奎因的话感到惊讶，他们在 1947 年总结了试图将数学建立在逻辑基础上的激进唯名论：
Logic alone does not suffice to establish an ontology of abstract objects. Nor should we be astonished by the words of Goodman and Quine, summarizing in 1947 the radical nominalism that was implicit in the attempt to base mathematics on logic:
我们不相信抽象的物体。没有人认为抽象实体——类、关系、命题等——存在于时空中；但我们想说的远不止这些。我们想完全消除它们的存在。6
We do not believe in abstract objects. Nobody supposes that abstract entities – classes, relations, propositions, etc. – exist in space-time; but we want to say something more than this. We want to do away with their existence altogether.6
仍然值得指出的是，“抽象对象”容易以多种形式存在，并容易体现在相对具体的实体中，在空间和时间中存在。这种情况至少取决于两个不同且不同的因素：在机器的物理空间和时间中发展的自动计算的存在，以及普遍认为数学实体类似于生物体的普遍信念，在某种程度上能够决定使我们能够研究和理解它们的具体条件。操作的可能性，受在某些条件下，在物理上作为二进制序列存在于计算器内存中的数字本身就代表了一个决定性的事实。约翰·冯·诺依曼能够观察到大规模的自动计算是如何从 40 年代开始随着第一台数字计算器的发展而发展起来的，它是如何由只有纯数学的抽象结构才能实现的时空计算组成的。能够详细说明。在新的微积分科学中，数学抽象和物理现实是密不可分的。
It is still worth specifying that ‘abstract objects’ are susceptible to existing in a variety of forms, and to becoming embodied in entities that are relatively concrete, with an existence in space and time. This circumstance depends on at least two distinct and different factors: the existence of an automatic calculation that develops in the physical space and time of a machine, and the widespread conviction that mathematical entities resemble living organisms, to the extent of being able to dictate the concrete conditions which permit us to study and understand them. The possibility of operating, subject to certain conditions, on numbers that exist physically as binary sequences in the memory of a calculator represents in itself a decisive fact. John von Neumann was able to observe how automatic calculation on a large scale, which had been progressing since the forties with the development of the first digital calculators, consisted of a calculation in space and time that was made possible by abstract structures that only pure mathematics was capable of elaborating. Mathematical abstraction and physical reality were inseparable in the new science of calculus.
当我们谈论“现实”时，我们正误入一个极其艰难的领域，事情可能会被颠覆：一个数字也可能是真实的，它并不像我们希望或期望的那样存在，但我们觉得有义务在会议过程中定义我们的需求或算法的需求。7无论如何，奎因本人在定义实数时指的是先前的理论，例如戴德金德和魏尔斯特拉斯的理论，而这些理论又得益于各种知识来源，至少可以追溯到第五卷中的比例理论。欧几里得元素. 而这一理论本身又因比欧几里得理论更古老的计算知识而得到加强。在衡量现实程度方面似乎具有决定性意义的是一系列理论和实践的经验和情况，它们通常起源于遥远的过去，因此几乎必须得出某个定义或设定一个特定的理论上，它希望以某种方式而不是另一种方式配置它。恩斯特·马赫 (Ernst Mach) 写了关于科学发现发生方式的非常有趣的文章：人们总是在未知与已知、分析与综合、发现与认识的不间断结合中重新审视已知事物。发现和发明通常由先前的历史支撑，并通过一系列促进他们并使他们变得必要的想法。
When we talk about ‘reality’ we are straying into an extremely arduous terrain where things may be upended: a number may also be real that does not exist as we would wish or expect but that we somehow feel obliged to define in the course of meeting our needs or those of the algorithm.7 In any case, Quine himself, in defining real numbers, was referring to preceding theories such as those of Dedekind and Weierstrass, which were in turn indebted to the various sources of knowledge stretching back at least to the theory of proportions in Book V of Euclid’s Elements. And this theory was enhanced, in its own turn, by a knowledge of computation that was much more ancient than Euclidean theories. That which appears to be decisive in gauging the degree of reality is a range of experiences and situations, both theoretical and practical, that often have their origins in the remote past and which make it almost essential to arrive at a certain definition or posit a specific theory, that wants it to be configured in a certain way and not in another. Ernst Mach wrote passages of great interest on the way that scientific discoveries occur: one always ends up revisiting that which was already known, in an uninterrupted combination of the unknown and the known, of analysis and synthesis, of discovery and recognition. Discovery and invention are usually sustained by a preceding history, and by a chain of ideas that have facilitated them and rendered them necessary.
为简化起见，我们可以先假设现实主义者是这样的人，他认为科学理论的命题要么是真要么是假，而使它们如此的人是我们外部的东西，不同于感官数据，不同于我们的语言的东西。和我们的想法。8那么，提出数字的真实性取决于物理世界的真实性这一概念是否合理？数学在物理学、经济学、工程、化学和信息技术中的无数应用形成了一个广度和功效令人印象深刻的知识体系，似乎能够根据某种自然主义哲学建立公式的现实，基于宇宙的想法完全由自然物体组成——位于时空中并受因果律支配的物体。宇宙是一本浩瀚的书，伽利略在《分析者》的著名段落中写道，用数学语言写的，如果不求助于三角形、圆形和其他类型的几何形式，就无法理解。但令人怀疑的是，仅这种情况才能使数学实体具有现实性。我们只能肯定，在物理理论方面不可能成为实在论者，在数学理论方面不可能成为唯名论者。9
To simplify things, one might begin by assuming that a realist is someone who contends that the propositions of a scientific theory are either true or false, and that which makes them so is something external to ourselves, something different from sensory data, from our language and our thought.8 So might it be plausible, then, to advance the notion that the reality of numbers depends on that of the physical world? The countless applications of mathematics to physics, economics, engineering, chemistry and information technology form a body of knowledge of impressive breadth and efficacy that seems capable of establishing the reality of formulas according to some kind of naturalistic philosophy, based on the idea that the universe consists exclusively of natural objects – objects situated in space-time and subject to causal laws. The universe is a vast book, wrote Galileo, in a famous passage in The Assayer, written in the language of mathematics and impossible to understand without resorting to triangles, circles and other types of geometrical form. But it is doubtful that this circumstance alone gives reality to mathematical entities; we can only affirm that it is not possible to be a realist with regard to physical theory and a nominalist with regard to the theory of mathematics.9
因此，关于物理学的哲学现实主义并不是建立数学现实的唯一途径。无论我们的大脑如何详细阐述计算，人们普遍认为，如果我们将其应用于现实世界，我们最终会发现自己面临的算法、公式和演示具有或多或少的可变轮廓，但能够无论我们如何决定条件和它们存在的方式，就像任性和固执的生物。这就是为什么数学家如此频繁地声称他们感到有义务识别某些实体而不是其他实体，他们真的很惊讶他们似乎来自另一个世界，或者至少来自我们感知能力、语言和心理框架之外的现实10 – 一个本身足以让我们重新考虑康德关于数学思维的先天特征的论点的情况，根据该论点，“理性只感知它根据自己的设计产生的东西”（纯粹理性批判，第二版前言，1787 年）。数学实体，正如许多科学家所设想的那样，尤其是查尔斯·赫米特所认为的那样，不是没有生命的、人造的结构，而是真实的、有自己的连贯性和意向性的生物，能够指导我们的研究和确定条件我们习惯上将这种自由和自主归因于我们自己的理性的支配。
Therefore, it is not a given that a philosophical realism regarding physics is the only way of establishing a mathematical reality. However much it is our mind that elaborates the calculations, it is widely recognized that, if we put aside their applications to the real world, we find ourselves ultimately faced with algorithms, formulas and demonstrations that have more or less variable contours but are capable of dictating regardless of us the conditions and modalities of their existence, like wilful and obstinate creatures. This is why mathematicians so frequently claim that they feel obliged to recognize certain entities and not others, to be genuinely astonished that they seem to come from another world, or at least from a reality which is outside of our perceptual capacity, language and mental framework10 – a circumstance which is enough in itself to make us reconsider Kant’s thesis on the a priori character of mathematical thinking, according to which ‘reason perceives only that which it produces itself according to its own design’ (Critique of Pure Reason, Preface to the second edition, 1787). Mathematical entities, as conceived by many scientists and as they were perceived to be by Charles Hermite in particular, are not lifeless, artificial constructions but real, living beings with their own kind of coherence and intentionality, capable of guiding our research and determining the conditions of that freedom and autonomy that we customarily attribute to the dictates of our own reason.
“要定义现实，”Simone Weil 指出，“没有什么比这更重要的了。” 她自己的第一个结论是：“真实是超越的；这是柏拉图的基本思想。11因此，出于至少两个原因，有必要从古代数学开始：数学实体的真实性问题自毕达哥拉斯哲学以及吠陀和美索不达米亚的计算以及现代微积分，从十六世纪至今，一直基于从古代科学发展而来的建筑。数字的真实性问题在古希腊被提出，至少是含蓄地试图理解数学实体与无限、与非存在之间的关系。，或无限。无限是缺席（stéresis ），纯粹的潜力——一切，为了存在和持久，都必须与无限的消极性对立起来。在希腊数学中，这是lógos的任务，在其中发现了现代数字的先驱。关系现象，以及由此产生的东西，是一种接近神灵的实体。不仅在希腊。正是在这种关系中人们需要找到数字的真实性：并非巧合的是，19 世纪的数学家建立了数字的本体论和比例概念（在欧几里得和前欧几里得计算科学中发现）的重新出现。算术连续统。正是这种情况赋予量词逻辑以价值，通过它我们可以肯定某个类的存在（存在量词），或者对于某些变量的每个值，必须满足允许一个数字被识别的条件（全称量词）——这种情况使伯特兰·罗素（Bertrand Russell）断言“现实感在逻辑中至关重要”。12这还不足以真正建立一个本体，但随后为弥补这种不足而采取的措施是对遥远时代已经众所周知的计算过程的理想延续——在中国、印度和美索不达米亚以及希腊。
‘To define reality,’ noted Simone Weil, ‘nothing is more important than that.’ And her own first conclusion was that: ‘The real is transcendent; and this is Plato’s essential idea.’11 Hence it is necessary, for at least two reasons, to begin with the mathematics of antiquity: the question of the reality of mathematical entities has been around since the time of Pythagorean philosophy as well as Vedic and Mesopotamian calculations – and modern calculus, from the sixteenth century to the present day, has been based on constructions developed from ancient science. The question of the reality of numbers was posed in ancient Greece, at least implicitly, in attempts to understand what relations mathematical entities have with the infinite, with the non-being of the ápeiron, or infinite. Infinity was absence (stéresis ), pure potentiality – and everything, to exist and to endure, had to oppose itself against the negativity of the limitless. This was, in Greek mathematics, the task of the lógos, of proportion, in which the precursors of modern numbers were found. The phenomenon of relation, and what derived from it, was an entity close to the gods. And not only in Greece. It was precisely in this relation that one needed to find the reality of numbers: not coincidentally, it was thanks to the reprising of concepts of relation and proportion (found in Euclid and in pre-Euclidean computational science) that the mathematicians of the nineteenth century established the ontology of numbers and the arithmetical continuum. It was this same circumstance that gave value to the logic of quantifiers through which we can affirm that a certain class exists (the existential quantifier), or that the conditions which allow a number to be identified must be satisfied for every value of certain variables (the universal quantifier) – a circumstance which allowed Bertrand Russell to assert that ‘the sense of reality is vital in logic’.12 This was not enough to really establish an ontology, but what was done subsequently to remedy such inadequacy is an ideal continuation of computational processes already well known in remote eras – in China, India and Mesopotamia as well as in Greece.
自最远古时代以来所采用的枚举过程往往旨在通过一种示范或神圣的姿态使事物——无论是人还是神——成为真实的，将它们介绍到世界舞台上，以使它们成为现实和可识别的。 ，拥有在空间和时间中存在的完全权利。枚举、人口普查、列表和目录在荷马和赫西奥德、埃斯库罗斯和希罗多德以及旧约中经常出现。枚举是lógos的特权，它暗示了一个选择和收集、聚合的过程通过不同的实体排列成一个整体。在毕达哥拉斯的传统中，数字本身就像空间中的一组点一样被分开，其顺序类似于军队或天空中的星星。因此，一个人计算的数字不仅是抽象的，而且是真实的和明显的。然而，随着不可通约量的存在，人们发现自然数，即arithmós不足以让那些今天我们称之为无理数的实体真正存在。量值之间的关系不能用整数之间的关系来表达。正如 Georg Cantor 在 19 世纪用对角线法继续证明的那样，存在从任何类型的枚举中逃脱的幻想实体。希腊数学试图间接地表示这些实体，通过由增长和衰减定律调节的一系列数字、关系和几何图形。现代数学将继续使用类似的序列来定义无理数。
The processes of enumeration employed since the most ancient times often had the purpose of making things – both men and gods – real by means of a demonstrative or divine gesture, introducing them on to the stage of the world in order to render them actual and recognizable, with a full right to exist in space and time. Enumerations, censuses, lists and catalogues recur frequently in Homer and Hesiod, in Aeschylus and Herodotus, as well as in the Old Testament. Enumeration was a prerogative of the lógos, which suggested a process of selection and collection, of aggregation ordered by means of different entities into a single whole. Numbers themselves, in the Pythagorean tradition, were separated like a set of points in space, with an order similar to that of armies or of stars in the heavens. Consequently, the numbers with which one counted were not only abstract but also real and manifest. Nevertheless, with the demonstration that incommensurable quantities exist it was discovered that the natural number, the arithmós, was not enough to give actual existence to those entities that today we call irrational numbers. There are relations between magnitudes which cannot be expressed as relations between whole numbers. As Georg Cantor would go on to demonstrate in the nineteenth century with the diagonal method, phantasmal entities exist that escape from any kind of enumeration. Greek mathematics attempted to represent these entities indirectly, by means of a sequence of numbers, relations and geometrical figures regulated by a law of growth and decay. Modern mathematics would go on to use analogous sequences to define irrational numbers.
2. 众神的数学
2. Mathematics of the Gods
很难说数学首先出现的原因和地点。如果我们倾听那些试图通过展示这种尝试的内在无意义来阻止任何寻找其起源的人的论点，这也许也是毫无意义的。在Human, All Too Human（第 249 段）中，尼采谴责了沉重的压力，以及系统地注视过去可能导致的深度厌倦。在他 不合时宜的沉思中，“论生命中历史的利弊”，他认为以最严格的方式守护和归档过去的古人的灵魂可能会成为收藏狂热的盲目狂热和压抑的好奇心的牺牲品这阻碍了对所有新的和重要的事物的任何冲动。在尼采之后，米歇尔·福柯解释说，任何知识体系的起源都是无数的，只寻求一个是没有意义的，好像它是一切的唯一来源。
It is hard to say why and where mathematics first arose. It is also pointless, perhaps, if we listen to the arguments of those who have sought to discourage any search for its origins by showing the intrinsic senselessness of such attempts. In Human, All Too Human ( par. 249), Nietzsche denounces the weighing down, the deep weariness that can result from a gaze cast systematically towards the past. In the second of his Untimely Meditations, ‘On the Uses and Disadvantages of History for Life’, he argues that the soul of antiquarian man, who guards and archives the past in the most exacting way, can fall victim to the blind frenzy of a collecting mania and to an oppressive curiosity that acts as an obstacle to any impulse towards all that is new and vital. After Nietzsche, Michel Foucault explains that the origins of any system of knowledge are numerous and that it makes no sense to seek only one, as if it were the sole source of everything.
在我看来，数学几乎不能免于这条规则：无论如何，有很多种数学——算术、数学物理、代数、几何、分析和统计——可以为不同的问题提供答案，遵守不同的标准和使用各种演示技术，即使在许多情况下对抽象数学结构的研究已经揭示了不同领域之间惊人的相似性，使我们能够瞥见统一知识的轮廓。随着时间的推移，数学理论经常改变方面，已经以多种方式构思和制定，并产生不同的结果，并扩展到其他理论。
It seems to me that mathematics is hardly exempt from this rule: in any case, there are many kinds of mathematics – arithmetic, mathematical physics, algebra, geometry, analysis and statistics – that provide answers to different questions, obey a diversity of criteria and use various techniques of demonstration, even if in many cases the study of abstract mathematical structures has revealed surprising affinities between disparate domains, allowing us to glimpse the outlines of a unified knowledge. Over the course of time mathematical theories have often changed aspect, have been conceived and formulated in many ways and with various outcomes, branching out into other theories.
数学已经无数次地诞生和重生，以至于让人想起塔木德中摩西和拉比阿基瓦的寓言中的传统概念。1根据寓言，摩西从上帝那里收到了用无限的花饰和皇冠装饰的文字写成的托拉。上帝预言，对于每一次繁荣，一个名为阿基瓦的尚未出生的人有一天会制定出无数的教义。摩西要求开悟，上帝允许他像施了魔法一样，坐在 Akiva 即将上课的教室第八排的学生旁边。迷惑不解，他完全不明白阿基瓦在解释什么。但是当学生们问拉比他是如何采取正确的方法来解决某个问题时，他感到放心——他回答说这是通过在西奈山上给摩西的一项法律。这包含了一个关于传统的深刻真理，关于根据更先进的公式不断重新调整古代知识，这些公式反过来又成为新的理解形式。
Mathematics has been born and reborn countless times, so much so in fact as to lead one to think of a concept of tradition like that in the fable of Moses and Rabbi Akiva in the Talmud.1 Moses, according to the fable, receives the Torah from God written in a script ornamented with infinite flourishes and crowns. For every flourish, God predicts, an as yet unborn person called Akiva will one day formulate innumerable doctrines. Moses asks to be enlightened, and God allows him, as if by magic, to sit alongside the pupils in the eighth row of the schoolroom in which Akiva is about to teach. Bewildered, he understands nothing of what Akiva is explaining. But he is reassured when the pupils ask the rabbi how he has taken the right route to tackle a certain question – and he replies that it was by way of a law given to Moses on Mount Sinai. This contains a deep truth about tradition, about the continuous readapting of ancient knowledge in the light of more advanced formulations that in turn become new forms of understanding. Mathematics is no exception to this process.
然而，我们可以确定一些基本主题，这些主题解释了数学是如何以及为什么随着时间的推移而采取它的形式，而不是其他的形式。在 19 世纪末，人们应该从哪些预设开始似乎终于很清楚了。理论上，数学源于智力的自由活动，源于不受外界环境限制的计算精神，以及简单的集合运算。最简单的智力操作，基本的并且显然不受矛盾影响，变成了将不同的实体（数字、函数、矩阵）组合成一个类的操作并尽可能在不离开该类的情况下尝试与这些实体进行操作。数学家们试图证明这个类是如何封闭的，2一种令人放心的财产，因为它可以防止在计算过程中不得不与不符合为该类别制定的规则的外国实体抗衡的可能性。有时你发现自己不得不离开某个指定的类别，例如代数方程的解，它可以有实数或复数，因此需要离开有理数领域。众所周知，复数的扩展源于对三次代数方程的解析进行的研究。紧随 Girolamo Cardano 之后的 Rafael Bombelli 是其首席建筑师。但是复数域本身对于加法和乘法运算是封闭的，在该域内，
We can, however, identify some foundational themes that explain how and why mathematics took the form that it did, over time, and not some other one instead. At the end of the nineteenth century it seemed finally to be clear from which presuppositions one should start. Mathematics, it was theorized, is born out of the free activity of the intellect, from a spirit of calculation unconditioned by extraneous circumstances, and by means of simple set operations. The simplest intellectual operation, fundamental and apparently immune to contradiction, became that of grouping together different entities (numbers, functions, matrices) into a single class and attempting to operate with these entities, if possible, without leaving that class. Mathematicians were seeking to demonstrate how the class operated in is closed,2 a reassuring property, because it prevents the possibility of having to contend, in the course of a calculation, with an alien entity that does not comply with the rules established for that class. Sometimes you find yourself obliged to leave a certain designated class, as in the case of solutions to an algebraic equation which can have real or complex numbers and consequently require a departure from the field of rational numbers. As is well known, the extension to complex numbers emerged from research conducted on the resolution of third-degree algebraic equations. Rafael Bombelli, following in the wake of Girolamo Cardano, was its principal architect. But the field of complex numbers is itself closed with regard to operations of addition and multiplication that generalize, within the field, the ordinary operations between rational or real numbers.
对越来越抽象领域的研究并没有削弱在 19 世纪逐渐形成的信念，即一切都可以追溯到简单的整数概念，而整数又可以基于更简单的概念：集合，或集合论。Weierstrass、Peano、Cantor、Dedekind 或 Veronese 的伟大结构都受到算术分析这一宏伟计划的启发，即使整个数学大厦至少在原则上源自整数、集合论和极限概念.
The study of ever more abstract domains did not undermine the conviction, gradually reached in the nineteenth century, that everything may be traced back to the simple concept of the whole number, while the whole number may in turn be based on the even simpler concept of the set, or set theory. The great constructions of Weierstrass, Peano, Cantor, Dedekind or Veronese were all inspired by the grand project of arithmetizing analysis, of making the entire mathematical edifice derive, at least in principle, from whole numbers, from set theory and from the concept of limit.
但是，有必要区分起源和派生。数学真的起源于 19 世纪科学所阐述的集合论的运算吗？如果我们坚持消息来源，那么答案只能是否定的。
It is necessary, however, to distinguish between origin and derivation. Did mathematics really originate from the operations of set theory such as those elaborated by nineteenth-century science? If we adhere to the sources, then the answer cannot be other than negative.
在古代美索不达米亚数学中，我们发现算术计算在种类上类似于现代程序：真正的算法avant la lettre。与建造火坛有关的吠陀几何包括研究几何图形之间的等价性，例如正方形的圆圈，要实现这一点需要精确的数值过程，例如近似
In ancient Mesopotamian mathematics we find arithmetical calculations similar in kind to modern procedures: genuine algorithms avant la lettre. Vedic geometry relating to the construction of fire altars includes the study of equivalence between geometric figures, such as the circling of the square, and to achieve this requires accurate numerical processes, such as the approximation of 2. For the Pythagoreans, numbers had a geometrical form, a circumstance which by definition demanded analysis of the relation between numbers and shapes and which would later lead – we may conjecture – to the discovery of incommensurable quantities. In Greece we find a geometry virtually connected to algebra and to the computatio which precedes the modern science of calculus. Its fundamental outlines can also be seen in ancient Chinese mathematics. Therefore, in general, one sees how, in different traditions, an enigmatic relation was established between number and geometrical figure – a problematic tension that in addition to giving rise to the study of incommensurability led to analysis of the concept of the infinite and of the structure of the continuum, as well as to the mathematical lógos, a science of relations as the foundation of an intelligible cosmos. And it was from this problematic tension that the first attempts were launched to give shape to important concepts such as incommensurability, effective constructability and approximation which would characterize all subsequent mathematics.
不可否认，计算在遥远文明的最早表现中揭示了与各种实践和知识体系的深刻关联。无论来源多么稀少，人们都可以合理地争辩说，数学和哲学、几何学和宗教、微积分和形而上学起源于一个伟大的、原始的、相互结合的起源。这种混合绝不会损害数学知识的特殊性；这并不意味着数学有其他用途或需要外部证明。如果有的话，情况正好相反：代数的公式和几何的构造具有无可置疑的特殊性，如此确凿的证据和如此清晰，可以说，它们不需要诉诸于自身之外的任何解释或证明。甚至逻辑也无法给出详尽的解释。计算似乎枯燥无味，与哲学或宗教领域格格不入，但这是一种误导性的印象：宗教与数学、形而上学与微积分、仪式行为和严格的思想似乎在一开始就结合成一个单一的，
It is impossible to deny that calculation reveals, in its earliest manifestations in remote civilizations, a profound affinity with a variety of practices and systems of knowledge. However scanty the sources may be, one can legitimately argue that mathematics and philosophy, geometry and religion, calculus and metaphysics descended from one great, original, reciprocally combined origin. This admixture in no way compromises the specificity of mathematical knowledge; it does not imply that mathematics serves some other purpose or is in need of external justifications. If anything, the opposite is the case: algebra’s formulas and geometry’s constructions enjoy an unquestionable specificity, so conclusively evidenced and of such clarity, so to speak, that they do not need to resort to any explanations or justifications outside of themselves. Not even logic is able to give an exhaustive explanation. Calculations appear dry and alien to the philosophical or religious domain, but this is a misleading impression: religion and mathematics, metaphysics and calculus, ritual action and exacting thought seem to combine, in the beginning, into a single, powerful nexus – a combination that one must seek to discern in the great designs of ancient cosmology, in the conceptions of the first philosophers, as well as in the computational strategies and most minute calculations of which Greek, Chinese and Babylonian mathematicians were so fond.
从资料中我们可以推断出第一个提出数学问题的人不是人而是神——或者至少是受神的启发的人。令人印象深刻的各种故事、解释和索引都证实了这一点。埃斯库罗斯的普罗米修斯是数字的发明者，也是第一个区分调节大时间周期的天文符号的人，而这需要数学关系理论。埃及神透特也有类似的特权。普罗米修斯的另一个自我，赫尔墨斯；尤其是帕拉梅德斯，他是半人马凯龙星的学生和特洛伊战争的英雄。柏拉图在Phaedrus (261 b-d)中提到了他，称他为“Eleatic”（埃利亚哲学学派的追随者）和他的逻辑能力与芝诺不相上下。对于高尔吉亚斯，我们欠帕拉梅德斯的防御（82 B 11a DK），它构成了史诗传统的一部分，替代了荷马史诗中的奥德修斯，在其中，奥德修斯的形象被稍微不那么崇高。正如扬布里库斯在《毕达哥拉斯的生活》（31）中断言的那样，在具有理性天赋的三个存在中，一个是上帝，另一个是人，第三个具有毕达哥拉斯的性格。毕达哥拉斯本人从埃及人和巴比伦人那里学到了一种从众神传来的测量艺术。巴门尼德从一位女神那里发现存在是独特而完美的。他的形而上学的愿景，与世界的数学概念兼容，可以与牛顿在他的《原理》中描述的绝对空间的概念相一致. 柏拉图断言，天神天王星授予我们数学，他通过恒星的运动表达了数字和关系的规律（Epinomis，977b）。
From the sources we can deduce that the first to pose mathematical problems were not men but gods – or at any rate men inspired by the gods. This is confirmed by an impressive variety of stories, explanations and concordances. The Prometheus of Aeschylus is an inventor of numbers and the first to distinguish the astronomical signs that regulate the great temporal cycles, and this would have required a mathematical theory of relations. Analogous prerogatives were attributed to the Egyptian god Thoth; to the alter ego of Prometheus, Hermes; and not least to Palamedes, the pupil of the centaur Chiron and hero of the Trojan War. Plato alludes to him in the Phaedrus (261 b–d), calls him ‘Eleatic’ (an adherent of the philosophical school of Elea) and attributes to him logical skills equal to those of Zeno. To Gorgias we owe the Defence of Palamedes (82 B 11a DK), which forms part of an epic tradition alternative to that of Homer in which Odysseus is cast in a somewhat less exalted light. As Iamblichus asserts in The Pythagorean Life (31), of the three beings with the gift of reason one is god, the other is man and the third has the character of Pythagoras. Pythagoras himself learned from the Egyptians and Babylonians an art of measurement transmitted from the gods. Parmenides discovered from a goddess that being is singular and perfect. His metaphysical vision, compatible with a mathematical conception of the world, could be aligned with the idea of absolute space that Newton would delineate in his Principia. Plato asserts that mathematics was granted to us by Uranus, the god of the heavens, who expressed the laws of numbers and of relations through the movements of the stars (Epinomis, 977b).
我们知道吠陀传统的诸神有一个特权：完全清楚和真实。正如查尔斯·马拉穆德（Charles Malamoud）所指出的，这也是他们的弱点，需要他们在与人类打交道时极度谨慎和胆怯。透明和真实意味着精确；因此，献给他们的祭坛需要准确和精确地设计和建造，仪式活动和仪式的执行也是如此。但是，出于自我保护的考虑，这种真实性可能会导致系统性的隐瞒：神明的名字被故意弄乱，以使他们无法接近，从而强化了他们的神秘和真相。根据Śatapatha Brahmana的说法，正是这种名称的扭曲，那神秘主义的起源。在这方面，我们想起赫西奥德的缪斯 ( Theogony , 27-8)所说的名言：“我们知道如何说出许多与真相相似的谎言，但我们也知道如何在选择时唱出真相。
We know that the gods of the Vedic tradition had a prerogative: to be utterly clear and truthful. And this, as Charles Malamoud notes, was also their weakness, necessitating an extreme caution and diffidence in their dealings with mankind. Transparency and veracity implied exactitude; hence the altars dedicated to them needed to be accurately and precisely designed and constructed, as did the performance of ritual actions and ceremonies. But this very veracity might lead, for self-protection, to a systematic concealment: the names of the deities were deliberately garbled in order to make them inaccessible, thus intensifying their mystery and truth. It is precisely with this distortion of the name, according to the Śatapatha Brāhmana, that mysticism originates. We are reminded in this regard of the famous words spoken by Hesiod’s Muses (Theogony, 27–8): ‘We know how to utter many lies that resemble the truth, but we also know how, when we choose, to sing that truth.’
也许这是让数学如此吸引人的原因之一，因为这是一门提出谜题的精确科学，通过基于计算、演示和算法的策略来解开深奥的问题。芝诺关于运动不可能性的悖论是最著名的谜团之一。但是深奥和神秘仍然符合毕达哥拉斯的原则，即“虚假不利于数字的本质”，正如菲洛劳斯（44 B 11 DK）的一个片段所言。最简单的几何图形，如直线、正方形、三角形或圆形，以最清晰和最容易理解的形式构成了最深奥的谜团，例如数量的不可通约性或无限或无穷小的数量的存在。对于毕达哥拉斯学派来说，形状的定义具有神学特征，正如普罗克鲁斯在他的欧几里得第一本书的“元素”（130）的评论，如果每个形状都与一个神相关联，这并不奇怪，这种习俗将通过几个世纪的传承，在连续的重新审视中，至少直到佐丹奴布鲁诺时代。
Perhaps this was one of the things that made mathematics so appealing, for here was an exact science that proposes enigmas, abstruse problems to be unlocked by way of stratagems based on calculations, demonstrations and algorithms. Zeno’s paradoxes concerning the impossibility of motion are among the most celebrated enigmas. But abstruseness and the enigmatical are still in accordance with the Pythagorean principle that ‘falsehood is inimical to the nature of number’, as a fragment of Philolaus (44 B 11 DK) has it. The simplest geometrical figures, such as the straight line, the square, the triangle or the circle, configured the most abstruse enigmas in the clearest and most accessible form, such as the incommensurability of quantities or the existence of infinite or infinitesimal quantities. For the Pythagoreans, the definition of shapes had a theological character, as Proclus reminds us in his Commentary on the First Book of Euclid’s ‘Elements’ (130), and it isn’t at all surprising if each shape was associated with a deity, a custom that would be transmitted down through the centuries, in successive revisitations, at least until the time of Giordano Bruno.
在希腊，数学问题的解决被转化为对众神的奉献，正如埃拉托色尼写给托勒密国王的一封信所证明的那样，这封信由阿斯卡隆的已故评论家阿斯卡隆的尤托西乌斯传给我们：他的建筑机制复制一个立方体的比例平均值变成了放置在神殿内的供品。在希腊语中，这构成了anáthema，一个与论文相关的术语（表示放置或建立某物的行为，也在司法环境中）。现在数学定理不针对任何东西除了证明论文之外，严格论证本身的发明可能是（在西蒙娜·威尔（Simone Weil）惊人的洞察力之后）对被认为是神圣现实的反映或体现的图像的集中注意力的结果。
In Greece the resolution of a mathematical problem was translated into a votive offering to the gods, as attested by a letter from Eratosthenes to King Ptolemy which has come down to us via Eutocius of Ascalon, the late commentator on Archimedes: his mechanism for the construction of the proportional mean to duplicate a cube became a votive offering placed within the deity’s shrine. This constituted, in Greek, the anáthema, a term related to thésis (denoting the action of placing or establishing something, also in a judicial context). Now mathematical theorems do not aim for anything other than to demonstrate a thesis, and the invention of rigorous demonstration itself could have been the result (following in the wake of a prodigious insight by Simone Weil) of intensified attention directed at images conceived to be reflections or embodiments of divine reality.
众神留给我们的是什么？数学是否应该感谢他们收集简单的轶事和传说，或者感谢真正知识的遗产？答案是毋庸置疑的。从众神提出的问题中产生了一种思维方式的基础，没有这种思维方式，现代数学将是不可想象的。
What is it that the gods have bequeathed to us? Does mathematics owe them a debt of gratitude for a simple collection of anecdotes and legends, or for a legacy of genuine knowledge? The answer is beyond doubt. From problems posed by the gods stems the basis of a way of thinking without which modern mathematics would be inconceivable.
我们可以确定神与人类秩序之间、诸神的基本命题与现代数学之间这种持久联系背后的原因，与其说是概括的轮廓，不如说是明确的目的或从可见世界中抽象出来的共享程序。这种联系比这更微妙，而且对于涉及最技术性和最秘密的计算操作，代数和分析仍然基于的基本范式，更加强大。决定性的联系发生在两个主要主题上：几何图形的增加方式和不同形状图形的等效性。
We can identify the reason behind this persistent link between divine and human orders, between the foundational propositions of the gods and modern mathematics, not so much in general outlines, in explicit ends or in a shared programme of abstraction from the visible world. The connection is more subtle than that, and all the stronger for involving the most technical and secret operations of calculation, the fundamental paradigms on which algebra and analysis are still based today. The decisive link occurs with respect to two main themes: the modality of increase of geometrical figures, and the equivalence of figures of different shape.
重要的参考资料可以在吠陀论文中找到，例如Śulvasūtra（s´ulva是绳索，工具，连同钉子，用于仪式测量）关于建造献给烈火的火坛。这些也是旨在阐明建立希腊数学的程序的起源和意义的小册子。这些小册子最重要的版本属于 Baudhāyana 学派、Āpastamba 学派和 Kātyāyana 学派，而这些来源首先回应了在两者之间建立对等性的需要。不同形状的祭坛。对等价的搜索导致构建与矩形相同面积的正方形，或与正方形相同面积的圆。从一个正方形的圆中，大概出现了相关的、更著名的问题：圆的平方。
The essential references can be found in Vedic treatises such as the Śulvasūtra (s´ulva is the rope, the tool, together with pegs, for ritual measurement) on the construction of fire altars dedicated to Agni. These are also tracts designed to clarify the origin and significance of procedures on which Greek mathematics was built. The most important versions of these tracts belong to the school of Baudhāyana, of Āpastamba and of Kātyāyana, and these sources respond above all to the need to establish equivalence between altars of different shapes. The search for equivalence leads to the construction of a square of the same area as a rectangle, or of a circle of the same area as a square. From the circling of a square emerged, presumably, the related and more famous problem of squaring the circle.
另一个要求是扩大祭坛，同时保持其形状不变。负责仪式的人必须设想，至少在原则上，用五层砖建造祭坛，祭坛的面积越来越大，表面等于 7½、8½、9½，最大为 101½ purusha平方，其中purusha是一个单位长度相当于一个人举起双臂的高度。这需要解决各种几何问题，例如扩大一个正方形，或者找到一个等于两个给定正方形之和或差的正方形，解决这些问题意味着了解毕达哥拉斯定理：长方形对角线上的和等于其边上的正方形之和。
The other requirement was to enlarge an altar while keeping its shape unchanged. Whoever was in charge of the ritual had to envisage, at least in principle, the construction in five layers of bricks of altars of increasing size with surfaces equal to 7½, 8½, 9½, up to 101½ purusha squared, where purusha was a unit of length equivalent to the height of a man with his arms raised. This entailed the solving of various geometrical problems, such as the enlargement of a square, or of finding a square equal to the sum or to the difference between two given squares, problems the resolution of which implied knowledge of Pythagoras’ theorem: the square made on the diagonal of a rectangle is equal to the sum of the squares made on its sides.
吠陀几何基于在随后的时代中描述的几何结构，在欧几里得的元素中。但是这两种处理方式并不总是具有相同的目标：在元素中我们发现了展示的严谨性，而在Śulvasūtra中，占主导地位的是几何图形的动态增长的想法和在经历变化时形状不变的原则在规模上。吠陀几何与希腊几何之间的距离尤其体现在二项式平方公式的表达式中，( a + h )² = a ² + 2 ah + h ²，这是现代发展的核心重要的代数等式分析。在里面Śulvasūtra of Āpastamba (III, 8) 3给出了特定大小的正方形扩大的某些公式：一根 1½ purusha的绳索通向一个面积相等的正方形到 2¼ purusa平方，因为二项式平方的公式变为 (1 + ½)² = 1 + 1 + ¼ = 2¼。换句话说，如果将边 1 的长度增加 ½，正方形的面积将增加 1¼。以类似的方式，如果在边 2 的正方形上加上 ½，则最终得到面积为 6¼ 的正方形，因为 (2 + ½)² = 4 + 2 + ¼ = 6¼。文本随后立即解释了更一般的规则：将增量 2 ah + h ² 添加到边a的正方形，当它经历增量h 时，由两个矩形 2 a h和一个正方形h ²组成一起形成一个晷针，4平方数，当应用于 a 边的正方形时，会产生一个面积更大的正方形 ( a + h )²。这是这个想法的核心：乔治·蒂博（George Thibaut），他在 19 世纪提供了吠陀关于建造火坛的论文的翻译和评论，在总结它时评论说，阿帕斯塔巴和卡提亚那提出了“重要的例子，说明了火坛的建造方式”。边长的增加或减少导致正方形的增加或减少。5但在《首尔瓦苏特拉》中，我们也发现了增减的普遍规律。
Vedic geometry is based on geometrical constructions which are described, in a subsequent era, in Euclid’s Elements. But the two treatments do not always share the same aims: in the Elements we find the rigour of demonstration, and in the Śulvasūtra what predominates is the idea of a dynamic growth of geometrical figures and the principle of the invariance of shapes when undergoing a change in scale. The distance between Vedic and Greek geometry manifests itself especially in the expression of the formula for the square of a binomial, (a + h )² = a² + 2ah + h², an algebraic equivalence of central importance for the development of modern analysis. In the Śulvasūtra of Āpastamba (III, 8)3 certain formulas are given for the enlargement of squares of particular size: a rope of 1½ purusha leads to a square with an area equal to 2¼ purusa squared, because the formula of the square of the binomial becomes (1 + ½)² = 1 + 1 + ¼ = 2¼. In other words, if you increase the length of the side 1 by ½, the area of the square increases by 1¼. In a similar way, if to the square of side 2 you add ½, you end up with a square of an area of 6¼ because (2 + ½)² = 4 + 2 + ¼ = 6¼. The text explains immediately afterwards the more general rule: the increment 2ah + h² to be added to the square of side a, when this undergoes the increase h, is made up of two rectangles, 2ah, and a square h² that together form a gnomon,4 the squared figure which, when applied to the square of side a, produces a square of greater area (a + h )². This is the nub of the idea: George Thibaut, who in the nineteenth century provided a translation of and commentary on the Vedic treatises on the construction of fire altars, when summarizing it remarked that Āpastamba and Kātyāyana advanced ‘significant examples that illustrate the way in which the increase or decrease in the length of a side causes the increase or diminution of the square’.5 But in the Śulvasūtra we also find the general rule of increase and of reduction.
不容忽视的一点是，欧几里得证明了关于二项式平方的公式的纯几何版本（Elements , II, 4）。欧几里得的论证所表现出的严谨性在韦达论文中是缺乏的，但欧几里得并没有提到公式的增量性质，这不仅是分析公式（从 16 世纪以后）的增量性质的开始，而且也是与它们相关联的计算过程。这种渐进的性质在Śulvasūtra中很明显。
Something that should not be overlooked is the fact that Euclid demonstrates a purely geometrical version of the formula regarding the square of the binomial (Elements, II, 4). The rigour manifested in Euclid’s demonstration is lacking in the Vedic treatises, but Euclid does not mention the incremental nature of the formula, which is the incipit not just of the incremental nature of formulas of analysis (from the sixteenth century onwards) but also of the computational procedures with which they are linked. This incremental nature is plainly explicit in the Śulvasūtra.
这些和其他几何结构，例如二项式的立方体，用于扩大几何立方体，已经构成了分析代数方程的主要工具——在文艺复兴时期和现代时期。在 16 世纪，复数的发现主要归功于 Rafael Bombelli 的工作，该工作源自对基于正方形或立方体（gnomonic）增量规则的二阶或三阶方程的研究。用晷针求解代数方程被证明是一个卓越的问题 ——在西方，亚历山大的席恩（欧几里得元素 的编辑）第一次面临) 在四世纪以及随后几个世纪由阿拉伯和意大利数学家提出。到 16 世纪末，法国数学家弗朗索瓦·维埃特（François Viète）开发了一种通用方法，用于数值求解任意次数的方程，该方程由正方形或立方体扩大规则的一种代数扩展组成。牛顿简化了维埃特的程序，17世纪末约瑟夫·拉夫森又发现了如何通过一个简单的迭代公式来表达牛顿的方法。6在同样的迭代公式中，我们今天找到了求解的基本模型，具有高效的自动程序、一般方程组和最小函数问题。
These and other geometrical constructions, such as the cube of the binomial for the enlargement of a geometric cube, have constituted the principal tool for the analysis of algebraic equations – in the Renaissance as in the modern era. In the sixteenth century the discovery of complex numbers, due largely to the work of Rafael Bombelli, derived from the study of second- or third-degree equations based on the rule of (gnomonic) incrementation of a square or a cube. The solving of algebraic equations by means of gnomons proved to be the problem par excellence – confronted for the first time, in the West, by Theon of Alexandria (an editor of Euclid’s Elements ) in the fourth century, and in subsequent centuries by Arab and Italian mathematicians. Towards the end of the sixteenth century the French mathematician François Viète developed a general method for solving numerically an equation of an arbitrary degree consisting of a kind of algebraic extension of the rule of enlargement of a square or a cube. Newton simplified Viète’s procedure, and at the end of the seventeenth century Joseph Raphson discovered in turn how to express Newton’s method through a simple iterative formula.6 In this very same iterative formula we find today the essential model for solving, with automatic procedures of great efficiency, general systems of equations and problems of minimum functions.
但是，依赖于扩大几何正方形的规则的不仅仅是单个数值程序。在这些相同的规则中，我们发现了推理形式的基础，这些推理形式导致了 17 世纪以来分析的发展，以及最近几年微积分系统的稳定化。作为一般规则，了解函数或整个计算过程对于变量的微小变化或增量如何表现至关重要。每当微小的变化、增量或干扰时，都会出现不稳定性变量的数量会导致函数值或计算过程的结果发生很大变化。
But it’s not only single numerical procedures that depend on the rules for enlarging a geometrical square. In those same rules we find the foundation of forms of reasoning that gave rise to the development of analysis from the seventeenth century onwards, as well as, in more recent years, to the stabilizing of systems of calculus. As a general rule, it is of utmost importance to know how a function or an entire process of calculation behaves for small variations in or increments of the variables. There is instability every time that small variations, increments or disturbances of the variables cause a big variation in the value of the function, or in the result of a process of calculation.
为了定义微积分的主要运算，即允许操纵新的无穷符号的代数自动性，莱布尼茨求助于诸如二项式平方的公式。7仅在此之前的几年，Bonaventura Cavalieri 在他的不可分割方法（用于确定几何形状的大小）中使用了相同的公式。8离散连续统问题的一些基本策略，用于将变量假定数值连续统的值的微分或积分问题转化为可以通过和和乘积的普通运算解决的算术性质的问题，基于公式它概括了二项式平方的表达式。在很大程度上，连续和离散之间的关系取决于这些完全相同的公式。
To define the principal operations of differential calculus, the algebraic automatism that permitted the manipulation of the new symbols of infinity, Leibniz resorted to formulas such as that for the square of a binomial.7 And only a few years prior to this, Bonaventura Cavalieri had used the same formulas for his method of the indivisibles (for determining the size of geometrical shapes).8 Some fundamental strategies for discretizing problems on the continuum, for translating differential or integral problems in which the variables assume the values of the numerical continuum into problems of an arithmetical nature that may be solved with the ordinary operations of sum and product, are based on formulas which generalize the expression of the square of the binomial. To a large extent, the relation between the continuous and the discrete depends on these very same formulas.
在Śulvasūtra中很难为供奉烈火的祭坛的几何形式找到精确的宗教含义，也无法在任何此类论文的基础上澄清这个问题，因为数学思想会继续采用，在随后的几个世纪中，我们熟悉的形式而不是另一种形式。然而，人们可以求助于更广泛的文献。精确的典故和数字、尺寸和几何运算的列表出现在通常被分配的起源比Śulvasūtra更古老的文本中，其中包括神话、仪式和形而上学类型的广泛注释。Taitirīya Samhitā(5, 4, 11) 包含一长串各种几何形状的祭坛，而在Śatapatha Brahmana中有不少段落提到祭坛的形状为一只猎鹰，预示着象征性的飞行。几何形状以purusha为单位，在最原始的版本中，祭坛的程式化身体类似于鸟，由七个正方形组成，其中四个构成中央躯干，两个代表翅膀，一个代表尾巴。据记载，元神 ṛṣi以方格的形式创造了七个不同的人，这些方格形成了一个单一的身体，最初，万物的创造者 Prajāpati 就是用这个身体制成的：宇宙的创造者和所有凡人以及不朽的存在者（Śatapatha Brahmana，VI，1, 1-3; X, 2, 3, 18）。
It isn’t easy to find in the Śulvasūtra a precise religious meaning for the geometrical forms of the altars dedicated to Agni, nor is it possible to clarify the issue on the basis of any such treatise, because mathematical thought would go on to adopt, in the centuries that followed, the form with which we are familiar and not another form. One can have recourse, however, to a much vaster literature. Precise allusions and lists of figures, dimensions and geometrical operations occur in texts that are usually assigned origins more ancient than those of the Śulvasūtra and which include extensive exegeses of a mythic, ritualistic and metaphysical kind. The Taittirīya Samhitā (5, 4, 11) contains a long inventory of altars of various geometrical shape, and quite a few passages in the Śatapatha Brāhmana refer to constructions of altars in the shape of a falcon, prefiguring symbolic flight. The geometric forms are measured in purusha, and in its most primitive version the stylized body of the altar resembling a bird was composed of seven squares, four of which make up the central trunk, with two for wings and one for a tail. It is written that the ṛṣi, the vital spirits, created seven different persons in the form of squares, and that the squares formed a single body, the body with which, in the beginning, Prajāpati, the creator of all things, was made: the creator of the universe and of all mortal as well as immortal beings (Śatapatha Brāhmana, VI, 1, 1–3; X, 2, 3, 18).
建造火坛的目的是什么？一开始是 Prajāpati 的传说，正如Śatapatha Brahmana (VI, 1, 1, 12-13) 中所描述的那样，由于他所负责的创造，他仍然破碎和分离。生命力从他身上逃脱，众神抛弃了他。但是Prajapati要求Agni将他重新组合在一起，而Agni正是由Prajapati重建的身体组成的祭坛。
For what purpose are fire altars built? One starts with the legend of Prajāpati, who, as related in the Śatapatha Brāhmana (VI, 1, 1, 12–13), remains shattered and disarticulated as a result of the very creation for which he is responsible. The life force escapes from him and the gods abandon him. But Prajāpati asks Agni to put him back together again, and Agni is precisely the altar that consists of the reconstructed body of Prajāpati.
图1
Figure 1
Śatapatha Brahmana (VI, 1, 2, 36) 解释说，赋予生命的神灵化身为鸟形，变成了 Prajāpati，并且“以 [祭坛的] 形式， Prajāpati 创造了众神”。新生的神又变得不朽。Brhadāranyaka Upanishad (I, 4, 6) 中的一段解释说，Prajāpati 设计了比他更优秀的神，尽管他自己只是凡人的本性，但他创造了本质上不朽的神。
The Śatapatha Brāhmana (VI, 1, 2, 36) explains that, assuming the form of a bird, the life-giving spirits became Prajāpati, and that ‘assuming that form [of the altar], Prajāpati created the gods’. The nascent gods in turn became immortal. A passage from the Brhadāranyaka Upanishad (I, 4, 6) explains that Prajāpati designed the deities that are superior to him and that, despite being himself of merely mortal nature, he generated that which is immortal in nature.
但是这种对神仙的超创造怎么可能通过凡人的工作而发生，我们又怎么能把宇宙的创造者想象成一个凡人呢？从《奥义书》的文本和《尚卡拉》的注释可以推断，般若可以从不同的角度来理解，并具有明显矛盾的特征。他在轮回之内和轮回之外平等地定位自己：虽然在休息，但他到处移动'（Śankara，Brhadāranyaka Upanishad，I，4，6）。如果沉浸在他所创造的形成中，般若菩提似乎是凡人、分裂、肢解、恐惧和孤独的牺牲品，因此，正如《布哈达兰尼卡奥义书》中所描述的那样(I, 4, 1)，要反恶，烧尽一切恶。这就是为什么他被称为普鲁沙；事实上，桑卡拉评论道，执着和无明的邪恶阻碍了对般若神性的觉悟。桑卡拉象征性地解释这个词的组成部分，指出普鲁沙指定他是第一个 ( pūrvam ) 烧毁 ( aushat ) 阻碍实现他真实本性的所有邪恶的人，并且同一位祈求者，在 Prajāpati 的重建和仪式的正确庆祝，获得与Purusha（Śankara，Brhadāranyaka Upanishad，I，4，1）的完美认同。
But how could this kind of hypercreation of an immortal happen through the work of a mortal, and how can we conceive of the creator of the universe as a mortal being? From the text of the Upanishads and the commentary of Śankara it can be deduced that Prajāpati can be understood from different angles and assumes apparently contradictory characteristics. He situates himself equally both within and beyond transmigratory becoming: ‘Although remaining sitting, he was far away; though at rest, he moves everywhere’ (Śankara, Brhadāranyaka Upanishad, I, 4, 6). If immersed in the becoming of which he is the author, Prajāpati appears mortal, divided, dismembered, prey to fear and loneliness and, consequently, as related in the Brhadāranyaka Upanishad (I, 4, 1), needing to counter evil, to burn away all evil. This is why he is called Purusha; in fact, Śankara comments, the evil of attachment and ignorance impedes the realization of the divine nature that is proper to Prajāpati. Interpreting symbolically the component parts of the word, Śankara points out that Purusha designates he who was the first ( pūrvam ) to burn (aushat ) all the evils that obstructed the realization of his true nature, and that the same supplicant, following Prajāpati’s reconstruction and the correct celebration of the rite, obtains perfect identification with Purusha (Śankara, Brhadāranyaka Upanishad, I, 4, 1).
火坛的建造需要应用我们在Śulvasūtra 中发现的精确规范，数学形式。但是正确连接的砖块达到了所需的形状，例如圆形或方形，它们本身就类似于神。每一块砖都代表着一个必须被召唤的神，Katha Upanishad宣称(I, 1, 15) 和 Mṛtyu 或死亡，它认识到实现神性的工具，并让自己摆脱因成为世界而表现出来的恐惧，展示了应该使用多少块砖，以及应该使用什么其他规格用于架设祭坛。与般若的情况一样，Mṛtyu 位于生成过程的内部和外部：在前一个位置，它是死亡、饥饿和恐惧的代名词；在后者中，它代表了一位救赎神，他教导火的力量作为净化和超越的手段。
The construction of the fire altar requires the application of precise norms, mathematical in kind, that we find in the Śulvasūtra. But the bricks which properly connected attain the required shapes, for example that of a circle or a square, are themselves akin to gods. Every brick represents a deity that must be invoked, declares the Katha Upanishad (I, 1, 15), and Mṛtyu, or Death, which is cognizant of the tools for achieving divine nature and freeing oneself of the fear caused by the manifestation of becoming in the world, shows how many bricks should be used and what the other specifications are for erecting the altar. As in the case of Prajāpati, so Mṛtyu is located both inside and outside the process of becoming: in the former location it is synonymous with death, hunger and fear; in the latter it represents a redemptive deity giving instruction on the power of fire as a means of purification and transcendence.
祭坛将内在和外在结合起来：Purusha 深深地隐藏在所有众生中，但我们没有感知到他，因为我们的感觉器官将我们向外投射到物质世界，使我们无法正确地向内看。9以几何形式构建的烈火是内部和外部、思想和自然的综合——现代科学将在 20 世纪初的基本知识危机中以戏剧性的方式经历的极性的数学解决方案。
The altar unites internal and external: Purusha is hidden deeply in all beings, but we do not perceive him because our sensory organs project us outwards to the material world and render us incapable of a properly inward perspective.9 Agni constructed in a geometrical form is the synthesis of the interior and exterior, of mind and nature – a mathematical solution to a polarity that modern science will experience, in a dramatic way, in the fundamental intellectual crises of the early twentieth century.
Śatapatha Brahmana (VI, 1, 3, 20)的一段段落帮助我们更好地理解火坛的特征与受制于增长过程的数字和几何形式的概念有关。那里规定祭坛必须在一年内竖立起来，并指出有些人更愿意确定它应该在两年内竖立起来。建筑工作和仪式诵读必须持续一年，在此期间，祭坛就像一颗注定要播种的种子发芽，'因为种下的种子是有生产力的；它在下面，不断变化和成长”。但是祭坛的生长必须尊重精确的数学比例：这一要求完全符合柏拉图关于数字级数作为符合phýsis的增长方式的愿景。在柏拉图的视野中，这种增长从数字 1 开始，被认为是似乎印在自然 ( phýsis ) 上的进程的生成原则。10
A passage of the Śatapatha Brāhmana (VI, 1, 3, 20) helps us to better understand what the character of the fire altar is in relation to an idea of numbers and of geometric form subject to a process of growth. It is specified there that the altar must be raised in the space of one year and noted that there are those who prefer to determine that it should be erected instead in two. The building work and the ritual recitation must last a year, during which the altar is like a seed destined to germinate, ‘because the seed that is planted is productive; it lies beneath, changing and growing’. But the growth of the altar must respect precise mathematical proportions: a requirement that accords perfectly with the Platonic vision of numerical progression as a modality of growth that conforms with phýsis. In the Platonic vision, such growth started from the number 1, conceived as the generating principle of the progression that is seemingly imprinted upon nature ( phýsis ).10
有了这样的假设，微积分的重要程序被开发出来，作为全面的数学知识遗产的一部分，我们仍然可以从中辨别出重建和发展 Prajāpati 的主要设备。
With presuppositions such as this, important procedures of calculus were developed, part of a comprehensive patrimony of mathematical knowledge in which we can still discern the principal devices for the reconstruction and growth of Prajāpati.
最后，我们可能会问自己：是什么让数学成为最适合重构 Prajāpati 的工具？我们可以冒险假设一个假设，回忆一下波伊修斯的算术论文中的一段话。波伊修斯解释说，自然界中存在的一切都归功于数学定律。四元素结合的联系，时间节奏的交替，天体的旋转：一切都可以用数字来描述。祭坛具有类似的宇宙学意义。但是数字本身，无论如何变化，因为存在不同种类的数字，总是保持相同的实体，因为它永远不会由与自身不同的实体组成。11在数字的概念中，我们发现了不变性原则，即在不与自己疏远的情况下采取所有可能形式的能力。这正是需要重组的神性。
We may ask ourselves, finally: what was it that made mathematics the most fitting instrument for the reconstruction of Prajāpati? We can hazard a hypothesis, recalling a passage from a treatise on arithmetic by Boethius. Everything that exists in nature, Boethius explained, owes its form to the laws of mathematics. The connections through which the four elements are combined, the alternation of temporal rhythms, the celestial revolutions: everything can be described by numbers. The altar had an analogous cosmological significance. But the number itself, however subject to variation, because different kinds of numbers exist, always retains the same substance since it is never composed of entities different from itself.11 In the notion of number we find the principle of immutability, the ability to assume all possible forms without ever becoming estranged from oneself. This, precisely, was the nature of the god that needed to be reassembled.
3. 数学和哲学公式
3. Mathematical and Philosophical Formulas
柏拉图赋予数学一种辩证的/论述的功能，也就是说，介于dóxa（被科学纠正但又不同于科学的思想或意见）和noûs或“智力”（哲学家国王的最高智慧（共和国， 511 d))。从此，形成了形而上学以及数学本身的整个过程的公式、表达式和思维方式。他们共享的术语包括lógos（话语，也包括数学关系）和spermatikòs lógos，斯多葛学派的开创性理由，产生的宇宙原理——也是通过巧妙的算法从中提取类似于无理数的数值关系的单位。Logistiké表示关系的算术计算，但也是用于指代深思熟虑的能力，类似于计算，嵌入灵魂的永恒部分。在柏拉图的《对话录》中，哲学表达“多与少”和“大与小”作为无限或无限的同义词经常出现（ápeiron ) 但也适用于表示用分数逼近无理数的方式：随着计算的进行，分数和无理数（不能表示为分数）之间的距离越来越小，而分子和分母分数变得越来越大。大约在 15 世纪中叶，新柏拉图主义的库萨尼古拉斯宣称，真理不允许多或少。他的话需要用speculo来阅读数学 ：真理处于中心位置，就像无限摆动的天平的枢轴一样，它允许人们通过更精细的重量分布更接近平衡点——但从未达到完美平衡。
Plato assigned to mathematics a dianoetic/discursive function, that is to say, an intermediate position between the dóxa (thought or opinion corrected by and yet distinct from science) and noûs, or ‘intellection’ (the supreme intelligence of philosopher kings (Republic, 511 d)). From this stem formulas, expressions and ways of thinking that have marked the entire course of metaphysics as well as mathematics itself. Among the terms they shared were lógos (discourse, but also mathematical relations) and spermatikòs lógos, the seminal reason of the Stoics, the cosmic principle of generation – but also the unit from which, by means of ingenious algorithms, numerical relations resembling irrational numbers were extracted. Logistiké indicated the arithmetical calculation of relations but was also the term used to refer to the faculty of deliberation, similar to calculation, embedded in the eternal part of the soul. The philosophical expressions ‘more and less’ and ‘great and small’ occur frequently in Plato’s Dialogues as synonyms for the indefinite or infinite (ápeiron ) but are also adapted to signify the way of approximating irrational numbers with fractions: as one advances with the calculation, the distance between the fraction and the irrational number (not representable as a fraction) becomes increasingly small, while the numerator and the denominator of the fraction become ever larger. Around the middle of the fifteenth century, the Neo-Platonist Nicholas of Cusa declared that truth does not permit of either the more or the less. His words need to be read in speculo mathematico : truth is in a central position, like the pivot of an infinitely oscillating balance scale that allows one to get closer to the point of equilibrium with ever more finessed distribution of weights – but without ever reaching perfect equilibrium.
灵魂的生命有一种数学形式。正如从亚里士多德的尼各马可伦理学中推断出的那样，对一个无理数的数学近似的过度（向上舍入）和缺陷（向下舍入）越来越多地与标志着我们道德的不断振荡相对应。生活——介于自我的过分和缺陷之间，就像 Scylla 和 Charybdis 一样，位于完美所在的中间道路的两侧。
The life of the soul had a mathematical form. As it seems legitimate to deduce from the Nicomachean Ethics of Aristotle, the more and the less, the excesses (rounding up) and defects (rounding down) of the mathematical approximations to an irrational number were the counterpart of the incessant oscillations that mark our moral life – between the excesses and defects that place themselves, like Scylla and Charybdis, either side of that middle way in which perfection resides.
可以扩展从数学中借用的哲学公式列表：antanaíresis是寻找两个量之间的除数的方法（欧几里得算法）；但斯多葛派使用了类似的术语来表示有远见的智慧。同一个词，antanaíresis，与德语aufheben，拉丁语tollere同源，黑格尔将继续用它把辩证法的两个互补的时刻，带走和保持结合起来，也可以被翻译为“解决”，解开回顾过去，回到第一原则。1欧几里得的prós ti ( Elements, V) 表达了“关系”的数学思想，而在托马斯·阿奎那（Thomas Aquinas）中，同样的思想成为把握创造神学概念意义的关键。柏拉图学说的假设之一包括在不同现象背景下运作的法则之间的结构类比或同源性——本质上相同且连贯。2在柏拉图之前，俄耳甫斯派和毕达哥拉斯派也有同样的假设。
The list of philosophical formulas borrowed from mathematics can be extended: antanaíresis is the method of searching for the divider between two quantities (the Euclidean algorithm); but the Stoics employed a similar term to denote far-seeing wisdom. The same term, antanaíresis, cognate with the German aufheben, the Latin tollere, with which Hegel would go on to unite the two complementary moments of dialectic, the taking away and the keeping, could also be rendered as ‘to resolve’, to untangle retrospectively, returning to first principles.1 So, too, Euclid’s prós ti (Elements, V) expresses the mathematical idea of ‘relation’, while in Thomas Aquinas the same idea becomes the key to grasping the meaning of the theological concept of creation. One of the assumptions of Platonic doctrine consists of the structural analogy, or homology, between laws – essentially identical and coherent – operating in diverse phenomenal contexts.2 The same presupposition was shared before Plato by the Orphics and the Pythagoreans.
这些规律似乎不依赖于我们判断的运用，而是依赖于一种内在的必然性。数学实体本身。这就是为什么这些公式似乎被赋予了封闭的力量：它们有自己的生命并且是自我延续的，与我们自己的期望相反，明确地决定了我们推理的形式。它们似乎属于我们思想之外的外部现实，也许这就是柏拉图不让任何不懂几何的人进入他的学院的原因。
These laws seem to depend not on the exercise of our judgement but on a kind of necessity intrinsic to the mathematical entities themselves. This is why the formulas seem to be charged with hermetic power: they have a life of their own and are self-perpetuating, in contrast to our own expectations, determining categorically the shape of our reasoning. They seem to belong to an external reality outside of our mind, and perhaps this was the reason why Plato would not admit into his Academy anyone who did not know geometry.
然而，随着时间的推移，同样的认识论公式（其准确性因与数学公式的相似性而得到加强和证明）要求其自身的自主性，表现出自己决定不可撤销的思想规律的雄心。由此产生了一种尴尬，这种尴尬可以在伽利略等现代科学家的书中找到，对他们来说，对数学的无知将使对自然世界的任何形式的研究都变得不可能。伽利略非常清楚柏拉图和毕达哥拉斯学派对人类智慧的钦佩到了将其视为“与神性的分享”的程度（关于两个主要世界系统的对话, I) 因为它能够理解数字的重要性——他自己也认为这一观点是正确的。但他同样意识到，毕达哥拉斯如此受人尊敬的奥秘无法用容易传达的公式或模糊的言辞来概括。如果毕达哥拉斯说 3 是万事万物的关键数字，因为每件事都是由开始、中间和结束决定的，所以不能把 3 变成一个完美的数字——一个比 2 或 4 更完美的数字。将认识论公式作为通用货币，作为可以在任何场合应用的原则，会使数字的秘密成为庸俗的琐事——而真理只是一种刻板印象。相反，真理需要通过推理和数学来研究示威。为了能够在一些认识论公式中看到适用于每个知识领域的结构，需要对计算有全面而深刻的理解，直到最小的细节——以及如何超越贝内代托·克罗齐继续嘲笑的知识，在Giambattista Vico的话，作为对“小天才”的研究。3
Nevertheless, over time the same epistemological formula, the veracity of which was reinforced and justified by affinity with the mathematical formula, demanded its own autonomy, manifesting the ambition to dictate by itself the irrevocable laws of thought. There emerged from this an embarrassment that can be found in the pages of modern scientists such as Galileo, for whom ignorance of mathematics would make any sort of investigation of the natural world impossible. Galileo knew full well that Plato and the Pythagoreans admired the human intellect to such a degree as to consider it as ‘partaking of the divine’ (Dialogue Concerning the Two Chief World Systems, I) on account of its ability to understand the significance of numbers – a view which he himself also held to be true. But he was equally aware that the mysteries for which Pythagoras was so venerated could not be summarized in easily communicated formulas or through vague rhetoric. If the Pythagoreans said that 3 is the key number to everything, because each thing is determined by a beginning, a middle and an end, one could not turn 3 into a perfect number – a number any more perfect than 2 or 4. The use of an epistemological formula as common currency, as a principle that could be applied on every occasion, would have made the secret of number a vulgar triviality – and truth a mere stereotype. Truth, instead, needed to be investigated by means of reasoning and mathematical demonstrations. To be able to see in some epistemological formula a structure applicable to every domain of knowledge required a full and deep understanding of calculations, down to their smallest details – and the knowledge of how to penetrate beyond what Benedetto Croce would go on to mock, in the words of Giambattista Vico, as the study of the ‘little geniuses’.3
然而，如果仔细考虑，开创现代科学的那种推理继续利用柏拉图和毕达哥拉斯公式中可识别的直觉。伽利略本人并不排除这种可能性。对动力学的发展所依赖的落体的研究使他求助于一个模型，该模型可以追溯到数学的起源，并且与连续统的性质密切相关：等价案例的存在在“主要”和“次要”之间（关于两个主要世界体系的对话，I）。
And yet, when considered carefully, the kind of reasoning that inaugurated modern science continued to make use of intuitions recognizable from Platonic and Pythagorean formulas. Galileo himself did not exclude this eventuality. The study of falling bodies, on which the development of dynamics depends, led him to resort to a model that can be traced back to the very origins of mathematics and is intimately linked to the nature of the continuum: the existence of a case of equivalence between ‘major’ and ‘minor’ (Dialogue Concerning the Two Chief World Systems, I).
基于连续性原则的类似思想范式似乎已经在美索不达米亚的古代算术中得到了概述，4并且它们可能启发了欧几里德的关系和比例概念（元素，V）：可公度的关系或不可公度的量，作为由一系列较小的数值关系加上一系列较大的数值关系组成的实体。数值求解方程的方法一直基于类似的范式，为了强调这一点，一种自最远古以来就广为人知的算法，即所谓的regula falsi或错误位置法，被阿拉伯数学家称为“规则”天平的平底锅'。5被追捧的未知值作为它的轴，并在其中找到了使用或多或少原理的数值近似的无限次振荡的平衡点。
Analogous paradigms of thought, based on the principle of continuity, seem to have already been outlined in the ancient arithmetic of Mesopotamia,4 and it is possible that they inspired the Euclidean concept of relation and proportion (Elements, V ): the relation of commensurable or incommensurable quantities as entities made up of a series of smaller numerical relations plus a series of larger numerical relations. The methods for solving an equation numerically have always been based on a similar paradigm, and to emphasize this an algorithm well known since the remotest antiquity, the so-called regula falsi, or false-position method, was called by Arab mathematicians the ‘rule of the pans of the scales’.5 The sought-after unknown value operated as its axis, and within it was found the point of equilibrium of an indefinite number of oscillations of numerical approximations using the principle of more or less.
一对不等臂的天平，原理阿基米德讨论了可比量级在与其重量成反比的距离上平衡的杠杆——阿基米德本人在启发式问题解决的阐述中使用了平衡标度的图像，甚至在最严格的演示之前。在抛物线平方的情况下，启发式方法具有机械特性，包括平衡图形的各个部分，从而完成抛物线P和内接三角形T（具有相同底和相同的高度），以及另一个图形的部分，其尺寸是已知的。最后证明了P的面积 是4 / 3吨。
The pair of scales with unequal arms, the principle of the lever by which commensurable magnitudes are balanced at distances in inverse proportion to their weight, was discussed by Archimedes – and Archimedes himself used the image of the balancing scale in the heuristic problem-solving expositions that preceded even the most rigorous demonstration. In the case of the squaring of a parabola, the heuristic method, which was mechanical in character, consisted of balancing the parts of a figure, which completed that of a segment of the parabola P and of an inscribed triangle T (with the same base and the same height), with the parts of another figure the measurements of which were known. It was demonstrated in the end that the area of P is 4/3 that of T.
欧几里德和阿基米德使用的穷举法或多或少是基于近似等价的方法。为了证明圆作为在它们的直径上构成的正方形相互站立（元素，XII，2），欧几里得参考了两个多边形序列，分别内接和外接一个圆，其面积与圆的面积相近。和缺陷。圆位于中间，对应于内部和外部多边形之间的理想但不可能的等价。通过增加多边形的边数逐渐接近这种理想等价的可能性允许证明论文。
The method of exhaustion used by Euclid and Archimedes was based on the more or less method that approximated an equivalence. To demonstrate that circles stand to one another as the squares constructed on their diameters (Elements, XII, 2), Euclid makes reference to two sequences of polygons, respectively inscribed within and circumscribing a circle the areas of which approximate those of the circle by excess and defect. The circle rests in the middle and corresponds to an ideal, yet impossible, equivalence between the inner and outer polygons. The possibility of getting incrementally closer to this ideal equivalence by increasing the number of sides of the polygon allows the thesis to be demonstrated.
天平图像的暗示性和有效性吸引了奥林匹斯山上的神灵，宙斯在特洛伊战争期间使用这种天平来决定命运，而雅典娜则在奥瑞斯特斯的审判期间行使正义，这两者都被认为是一个中心对比结果之间的决策作用。
The suggestiveness and efficacy of the image of the balancing scale appealed to the very divinities on Olympus, where Zeus used such a balance to decide fates during the Trojan War, and Athena to exercise justice during the trial of Orestes, both according to it a central decision-making role between contrasting outcomes.
用于求解方程组或计算方程最小值的大规模现代算法函数，仍然基于启发生活在美索不达米亚（约公元前 1800-1700年）或吠陀印度等古代文明中的数学家的想法。Otto Neugebauer 和 Abraham Sachs 分析的可追溯到公元前7289 年的巴比伦石板有一个正方形，它的对角线和一些数字似乎表明美索不达米亚知道通过过量和缺陷的精确近似值
Large-scale modern algorithms for solving a system of equations, or for calculating the minimum value of a function, are still based on the same idea that inspired mathematicians who lived in ancient civilizations such as that of Mesopotamia (around 1800–1700BC) or Vedic India. A Babylonian tablet dating to 7289BC analysed by Otto Neugebauer and Abraham Sachs has a square with its diagonals and some numbers which seem to indicate knowledge in Mesopotamia of accurate approximations by excess and defect of 2. This knowledge was subsequently inherited, Neugebauer speculated, by Indian mathematicians.
伽利略用来研究物体运动的连续性定律，在 17 世纪引发了现代科学方法的革命，它已经在美索不达米亚的计算和吠陀数学中得到了预示。如果没有对这些计算的及时理解，就不会有关于古人的知识，甚至没有关于现代科学和上世纪微积分的知识——甚至可能连从古代毕达哥拉斯主义到柏拉图主义在连续几个世纪中发展起来的形而上学在西方。
The law of continuity used by Galileo to study the motion of bodies, from which in the seventeenth century the modern revolution in scientific method arose, can be found already prefigured in Mesopotamian calculations and Vedic mathematics. Without the timely understanding of those calculations, there would have been no knowledge of the ancients, or indeed of modern science and the calculus of the last century – and perhaps not even the metaphysics that from ancient Pythagoreanism, through the Platonists, developed throughout successive centuries in the West.
所有这一切的原因都可以在简单的操作中找到
The reason for all of this may be found in the simple operations that it is assumed made it possible for a scribe to approximate 2. These operations seem to be dictated, as Neugebauer and Sachs conjectured, by the idea of constructing an indefinite sequence of increasingly small intervals, one inside the other, and with the extremities defined by two rational numbers, each one of which contains 2. The irrational number 2 would then be configured, from the time of Mesopotamian calculus in antiquity, as a mathematical entity defined by the sequence of intervals that contain it. Similar strategies for approximating relationships between incommensurable quantities, through relations between whole numbers, would go on to be developed in Greece – and the definition of a real number, at the end of the nineteenth century, would rely on similar models of calculation to those used in Mesopotamia. Neugebauer seems to be perfectly aware of this remote derivation. We should note finally that the hypothetical algorithm that would have allowed the scribe to approximate 2 is a specific example of a general method, refined between the sixteenth and eighteenth centuries, for solving numerically an algebraic equation of any degree – a method which not only serves to approximate an irrational number but is also a sort of clavis universalis, or universal key, a fundamental strategy of mathematical analysis applicable throughout history.6
4. 增长与减少，数量与性质
4. Growth and Decrease, Number and Nature
事后看来，吠陀祭坛的几何形状需要对数字和几何图形之间发生的关系进行研究。直线-数的关系与不可通约的量和代数方程的解有关，而曲线-直线的关系将导致穷举法、无穷小计算和超越数如e或π的研究。1在这两种情况下，数学都不得不面对无限的非现实性，它有可能颠覆我们与世界的所有联系。
The geometry of Vedic altars would have required, with hindsight, a study of the relation that occurs between numbers and geometrical figures. The relation straight line–number was connected to incommensurable magnitudes and to the solution of algebraic equations, while the relation curve–straight line would have led to the methods of exhaustion, to infinitesimal calculation and the study of transcendental numbers such as e or π.1 In both cases mathematics was obliged to confront the irreality of the infinite, which had the potential to subvert all of our contacts with the world.
一个决定性的情况随之而来：众神自己要求描述数量如何变化，它们如何增长以及它们如何减少。根据亚里士多德的公式（形而上学，983 a 27-8），增长和减少使事物本身的本质受到怀疑，即tò tí ên eînai成为它本来的样子'。2从所有意图和目的来看，形而上学的基础都隐含地要求解释任何给定的实体，尽管可能会发生变化，但仍能保持其基本特征。实体需要能够被识别，而这种识别代表了其定义所必需的预设（horismós，lógos ）。
A decisive circumstance ensued: it fell to the gods themselves to demand a description of how quantities vary, how they grow and how they decrease. Growth and decrease threw into doubt the very essence of things themselves, the tò tí ên eînai, according to the Aristotelian formula (Metaphysics, 983 a 27–8), the quod quid erat esse : ‘the fact, for a thing, to continue to be that which it was’.2 To all intents and purposes, the very foundation of metaphysics implicitly required an explanation of how any given entity, although subject to change, could yet preserve its essential characteristics. The entity needs to be capable of being recognized, and this recognition represents the presupposition necessary for its definition (horismós, lógos ).
增长问题在古代文化中很普遍，并且具有可以追溯到太阳和月亮运动的宇宙学方面。对于 Orphics，正如 Proclus 在他对柏拉图的“共和国”（II, 58, 10-11）：“整个年周期分为增长和下降”。星光体，尤其是月亮，控制着地球上生命的增长和减少之间的类似振荡：月亮影响着所有的月下生物，并激发了一切，这要归功于它的力量、增长和减少（对柏拉图“蒂迈欧斯”的评论） , 二, 87, 20–28, 12)。
The question of growth was widespread in the cultures of antiquity and had cosmological aspects that could be traced back to the movements of the sun and the moon. For the Orphics, as Proclus relates in his Commentary on the ‘Republic’ of Plato (II, 58, 10–11): ‘the entire annual cycle is divided between growth and decrease’. And the astral bodies, especially the moon, governed an analogous oscillation between growth and decrease in life on Earth: the moon influences all sublunary beings and provokes in everything, thanks to its power, growth and decrease (Commentary on the ‘Timaeus’ of Plato, II, 87, 20–28, 12).
成长的概念被包含在更广泛的主题中，即在成为过程中的变化和身份丧失。西西里诗人和剧作家 Epicharmus 在公元前5 世纪写作，在致力于毕达哥拉斯学说的第一批证词中谈到了这一点。Iamblichus 讲述了剧作家如何避免公开进行哲学思考，而是更喜欢通过以消遣和戏剧表现形式（23 A 4 DK）掩盖毕达哥拉斯的思想来呈现它们。Epicharmus 是最早在喜剧中提出这样一个问题的人之一，即希腊人称之为állo的增长现象如何存在。 ——那总是其他的、不同的、类似于无形和变化的物质、不稳定和不真实的流动。他的角色，就像后来在阿里斯托芬的《云》中年长的斯特雷西德斯的案例一样，被要求偿还几年前所欠的债务——但否认他们是债务人，因为他们不再是以前的人了：这一论点令人耳目一新诡辩的。一个片段更明确地解释了这个问题（23 B 2 DK）：
The notion of growth was included within the broader theme of change and loss of identity during the course of becoming. Writing in the fifth century BC, the Sicilian poet and dramatist Epicharmus speaks of this in one of the first testimonies devoted to the doctrine of Pythagoras. Iamblichus relates how the dramatist refrained from philosophizing openly, preferring instead to present Pythagoras’ ideas by veiling them in the form of diversions, of theatrical representations (23 A 4 DK). Epicharmus was among the first to raise, within comedy, the question of how present in the phenomenon of growth is what the Greeks called állo – that which is always other, different, similar to formless and changing matter, an unstable and unreal flow. His characters, just as later in the case of the elderly Strepsiades in Aristophanes’ The Clouds, are required to pay a debt incurred years previously – but deny being debtors because they are no longer the same people they had been before: an argument that smacks of sophistry. A fragment explains the issue more explicitly (23 B 2 DK):
- 所以现在也考虑一下人类：事实上，一个人长大了，另一个反而衰落了：
- 简而言之，一切都在不断变化。
- 现在，本质上必须改变的东西，永远不会保持不变，
- 已经和原来不一样了。
- 事实上，你和我一样，都是昨天而不是现在，
- 明天又会不一样：永远不一样，按照同样的规律。
可变性是给定的，但它可以根据法律（lógos ）发生。我们可以推测，这个定律具有数学意义，它完全取决于关系的概念，通过这种关系，不同的事物相互关联——或者是同一事物的不同部分。我们从亚里士多德那里得到了第一个证实，他肯定（在气象学，379 b-380 a）“在某种关系（lógos ）持续的所有时间内，事物的性质（ phýsis ）保持不变”。从毕达哥拉斯和Śulvasūtra 开始, 数学一直渴望在变化过程中保持这种持续性或不变性——就像几个世纪以来从数学中汲取灵感的柏拉图哲学一样。
Mutability is a given, but it can happen according to a law (lógos ). We can conjecture that this law has a mathematical meaning and that it depends precisely on the notion of relation, by means of which diverse things are interrelated – or are different parts of the same thing. We receive a first confirmation of this from Aristotle, who affirmed (in Meteorology, 379 b–380 a) that ‘for all the time that a certain relation (lógos ) lasts, the nature ( phýsis ) of a thing remains unchanged’. Starting with Pythagoras and with the Śulvasūtra, mathematics has always aspired to this duration or invariability during the process of change – just as, over the course of the centuries, the Platonic philosophy that took its inspiration from mathematics has.
增长可能是无限的，也可能是减少。如果没有限制，随着增长，我们朝着无限大的方向移动，随着减少朝着无限小的方向移动——在这两种情况下，最终占主导地位的是ápeiron的非存在，实体将自己从我们的能力中移除想象它，变得无法定义。但是，导致这种毁灭的不仅是无限。最近，我们已经确定，由于数字的异常和不受控制的增长，数学实体在有限的计算过程中也接近于不可定义性。
Growth may be unlimited, and decrease may be as well. If there is no limit, with growth we move in the direction of the infinitely large, and with decrease towards the infinitely small – and in both cases what ultimately prevails is the non-being of the ápeiron, with the entity removing itself from our capacity to conceive of it, becoming indefinable. But it is not only the infinite that entails this kind of annihilation. In more recent times we have ascertained that a mathematical entity also verges on undefinability during the finite process of calculation, due to an abnormal and uncontrolled growth of numbers.
如果我们转向希腊，与数量增长有关的所有事情的一个关键概念是phýsis。但是，如果没有进一步的限定，将phýsis简单地翻译为“自然”是一种误导。亚里士多德（in Metaphysics , 1014 b 16–1015 a 19；Physics 193 a 28–193 b 21）走得更远，考察了各种可能的含义，并指出自然必须首先与生长和生成的观念联系起来. 亚里士多德断言，自然是“万物生长的第一个内在元素”（形而上学，1014 b 17-18）；以及“它直接存在于其中的事物的运动和静止的原理和原因，它本身而不是偶然的”（物理学，192 b 20）。
If we turn to Greece, a key concept regarding everything to do with the growth of quantities was that of phýsis. But it is misleading to translate phýsis as simply ‘nature’, without further qualification. Aristotle (in Metaphysics, 1014 b 16–1015 a 19; Physics 193 a 28–193 b 21) goes a good deal further, surveying various possible meanings, and specifies that nature must connect, above all, with the ideas of growth and generation. Nature, Aristotle asserts, is ‘the first immanent element from which follows all that grows’ (Metaphysics, 1014 b 17–18); and also ‘a principle and a cause of motion and rest for the thing in which it immediately resides, for itself and not by accident’ (Physics, 192 b 20).
亚里士多德的自然概念似乎沿着两条不同的轨道移动：一条可以追溯到先前的唯物主义理论，另一条与受形式 ( morphé ) 和lógos 支配的运动观念保持一致。关于其中的第一个，亚里士多德提醒我们，一些哲学家认为其产品的本质和现实 ( ousía ) 3存在于它们的物质性中 ( Physics , 193 a 10-12)。亚里士多德用来表示这种物质性的名字是质子 arrýthmiston，它是事物的第一个构成元素，仍然没有形式，就像床架的木头或雕像的青铜一样。阿瑞斯米斯顿 - 比例差，没有节奏 - 是rythmós的否定，这个术语将运动和形式这两个概念结合起来，在假设“节奏”的意思之前，它可以在更一般的意义上表示运动所采用的形式, 流体和需要修改 (Chantraine, Benveniste)。
The Aristotelian concept of nature appears to move along two distinct tracks: one which may refer back to prior materialistic theories, the other in keeping with an idea of motility governed by form (morphé ) and lógos. Regarding the first of these, Aristotle reminds us that some philosophers contend that the nature and the reality (ousía )3 of its products reside in their materiality (Physics, 193 a 10–12). The name Aristotle uses to denote this materiality is prôton arrýthmiston, the first constitutive element of a thing, still devoid of form, like the wood of a bed frame or the bronze of a statue. Arrýthmiston – badly proportioned, without rhythm – is the negation of rythmós, the term which unites the two notions of movement and form and which, before assuming the meaning ‘rhythm’, could mean in a more general sense the form adopted by that which is mobile, fluid and subject to modification (Chantraine, Benveniste).
因此，亚里士多德强调phýsis的含义是事物起源的根源，这些事物仍然没有形式，并且无法通过自身进行改变，从而使它们超越自身的虚拟性或潜力（dýnamis ）。他解释说，在hýle或拉丁语silva的意义上，自然可以被理解为物质而不是形式。但是亚里士多德也反对这种另一种观点，根据这种观点，我们可以构想一种基于形式本身（morphé ）的自然观念，以及它根据lógos（tò eîdos tò katà tòn lógon ）表现出来的方面。
Aristotle emphasizes, then, the meaning of phýsis as at the root of the provenance of things that are still devoid of form and incapable, by themselves, of undergoing a change taking them beyond their own virtuality or potential (dýnamis ). Nature may be understood, he explains, in the sense of hýle, or the Latin silva, as being matter as opposed to form. But Aristotle also opposes to this another point of view, according to which we can conceive an idea of nature based on form itself (morphé ), and on the aspect it manifests according to the lógos (tò eîdos tò katà tòn lógon ).
这两种立场的比较代表了我们关于自然的观念历史上的一个突出点。海德格尔也强调了这一点，他评论说，将自然视为物质，具有从运动中剥离与不变性和永久性有关的每一个特征的效果。如果构成事物的物质是水、火和土——如果构成元素只是原子——那么phýsis的流动和运动就属于单纯的消逝和不一致的标志。为了与这一愿景保持一致，“所有具有运动特征、每一次变化和每一个可变状态的事物 [ rythmós ]，最终出现在实体偶然发生的情况中；作为一种不稳定的东西，运动是一个非实体”。4
The comparison of these two positions represents a salient point in the history of our ideas about nature. Heidegger highlighted as much, commenting that the vision of nature as hýle, as matter, had the effect of divesting from movement every characteristic pertaining to invariability and permanence. If the matter of which things are made is water, fire and earth – and if the constitutive elements are only atoms – then the flux and the movement of phýsis fall under the sign of mere evanescence and inconsistency. In keeping with this vision, ‘everything that has the character of movement, every alteration and every variable state [rythmós ], ends up among what only occurs accidentally to the entity; being something which is unstable, movement is a non-entity’.4
那么，另一方面，当被精确地视为运动和形式的成长、节奏、符合逻辑的图形或外观时，自然是什么样的呢 ？海德格尔本人指出，在数学家的语言中，lógos意味着类似于关系和融洽的东西，而lógos是 légein 的实体，首先意味着使用辨别过程进行收集，因此，我们可以添加，选择单个离散的操作，以使只有一个，与数字相连并与数字结合——通过数字带来“分散事物的统一”。5数学如何处理数字和关系？关系将两个独立的量结合起来，在寻找两者共同的量度的过程中，使它们成为一体。以类似的方式，方程将同一公式中的不同量相关联。数学的集合的概念旨在将不同的东西收集到一个集合中；递归方法将各种运算（例如，数字之间所有可能的加法）集合在一个运算概念中，即sum。
So what, on the other hand, was nature like when seen instead precisely as movement and the growth of form, as rhythmós, as a figure or appearance conforming to the lógos ? Heidegger himself pointed out that, in the language of mathematicians, lógos means something akin to relationship and rapport, that lógos is the substantive of légein, meaning above all to collect using a process of discrimination, and hence, we could add, to select single discrete operations in order to make only one, articulated and united with numbers – to bring through number ‘a unity to scattered things’.5 How does mathematics act with numbers and relations? Relation unites two quantities that are separate, making them one, as it were, in the process of looking for the measure that is common to both. In an analogous way, an equation correlates different quantities in the same formula; the mathematical concept of the set aims to gather different things into a single collection; the method of recursion collects together a variety of operations – for instance, all the possible additions between numbers – in a single concept of operation, the sum.
因此，亚里士多德的话语间接地提到了变形现象，提到了在变化过程中转变身份的问题，以及逃避生成的偶然性的可能性——通过相对稳定的配置，最终逻辑和形态占主导地位。每件事都变成了一个摇摆不定的形象，毕达哥拉斯在奥维德的变形记(XV, 178) 中宣称：' cuncta fluunt omnisque vagans formatur imago '。但是，如果我们看一下数字和几何图形，就会很清楚如何通过关系来调节变化过程，以及如何在不改变其基本形式的情况下增加数量。只有在形式保持稳定的情况下，才有可能在成长过程中得到认可：以前的东西现在仍然存在。因此，亚里士多德的论点变得更加重要，其中关系的持久性（lógos ）保证了特定事物的性质（ phýsis ）的恒定性。
Hence Aristotelian discourse alludes indirectly to the phenomenon of metamorphosis, to the question of transforming identity in the process of change, and the possibility of escaping the accidental nature of becoming – by way of relatively stable configurations, where lógos and morphé ultimately prevail. Each thing becomes a wavering image, declares Pythagoras in Ovid’s Metamorphoses (XV, 178): ‘cuncta fluunt omnisque vagans formatur imago ’. But if we look at numbers and geometrical figures, it becomes clear how the process of change can be regulated by a relation, and how a quantity might be grown without altering its basic form. Only if the form remains stable is there the possibility of recognition in the process of growth: that which was before continues to be now. The Aristotelian thesis becomes consequently even more significant, whereby the permanence of a relation (lógos ) guarantees the constancy of the nature ( phýsis ) of a particular thing.
柏拉图的一段话揭示了形式的稳定性在多大程度上被认为是神圣的特权（共和国，380 d）：
A passage from Plato reveals just how much stability of form was to be considered a divine prerogative (Republic, 380 d):
那么，对于这条其他法律 [ nómos ]，我们能说些什么呢？你是否认为神是一种魔术师，他能够为了愉悦的欺骗我们，一次以一种特定的形式出现在我们面前，另一次以不同的形式出现在我们面前，或者改变他的面貌[ eîdos ]成一群不同的人物，目的是欺骗我们认为他是这样的？
What can one say, then, of this other law [nómos ]? Do you think perhaps that the god is a kind of magician who is capable, for the pleasure of deceiving us, of appearing before us at one time in a particular form and at another in a different one, or of changing his aspect [eîdos ] into a crowd of different figures with the aim of deceiving us into thinking that he is like this?
“将他的面貌变成一群不同的人物” ( alláttonta tò autoû eîdos eis pollàs morphás ) 渲染了变形的戏剧，并将其呈现为神性面孔的反复无常和多变的对立面。与此相反，神似乎是可识别的和真实的，就像在山峰上阿尔忒弥斯的顿悟一样，荷马在《奥德赛》中谈到了这一点（VI，102-9）。6所以eîdos，期待已久的愿景与现代经验主义者难以捉摸和幻想的想法相去甚远，它不包含欺骗或魔术，而是包含准确和真实的经验——因此不能简单地分解为一连串不连贯和不和谐的图像。一切都在phýsis的流动中成长、减少和转变，但神显脸色不变。也许正是出于这个关键原因，在不同的几何图形之间经常寻求对等关系，而众神的祭坛——希腊的阿波罗祭坛，就像印度的阿格尼祭坛一样——必须能够在规模上改变，同时保持它们的形式不变。数学承担了使这种不变性成为可能的任务，而这种不变性是公认的神圣形象的先决条件。不断变形的海洋老人普罗透斯也不例外，因为他通过对比真实性原则来暗示：他从模糊的多个海流中出现，一个推论他的易变性，其强度和不稳定程度与人们在存在之海中可以抓住的相同，在短暂的闪光中，7荷马分配给变形虫，此外，数字和数字推算（ légein ）的知识（奥德赛，IV，411-13）。
That ‘changing his aspect into a crowd of different figures’ (alláttonta tò autoû eîdos eis pollàs morphás ) renders the drama of metamorphosis and presents it as the inconstant and changeable antithesis of the face of the divine. In opposition to this, the god appears to be recognizable and real, as in the epiphanies of Artemis on the summits of mountains, of which Homer speaks in the Odyssey (VI, 102–9).6 So the eîdos, the long-awaited vision, far from resembling the elusive and phantasmal idea of modern empiricists, does not consist of deception or magic but of exact and truthful experience – and therefore cannot simply disintegrate into a flux of incoherent and discordant images. Everything grows, diminishes and is transformed in the flow of phýsis, but the god shows his face to be unchanging. Perhaps it is for this crucial reason that relations of equivalence were frequently sought among different geometrical figures, and the altars of the gods – of Apollo in Greece, just as of Agni in India – had to be capable of being altered in scale while maintaining their form unaltered. Mathematics took on the task of making possible this invariability that was a prerequisite of a recognizably divine figure. Proteus, the Old Man of the Sea, who was subject to continual metamorphosis, was not an exception to the rule, because he alludes by contrast to a principle of veracity: he emerges here and there from the indistinct plurality of marine currents, a corollary of his mutability, with the same intensity and precariousness with which one may grasp in the sea of existence, in a brief flash, the immutability of the Platonic Good.7 Homer assigned to Proteus, moreover, the knowledge of numbers and of numerical reckoning (légein ) (Odyssey, IV, 411–13).
但是增长和减少不仅是一般过程，而且可以指由初始核产生的膨胀和收缩现象，由原始在此过程中保持不变的形式。这种情况使得将哲学与数学联系起来成为可能。这也催生了一种可以持续数百年的放大（或缩小）技术——即使在最先进的代数计算中也可以识别这种技术。
But growth and decrease were not only generic processes and could refer to phenomena of expansion and contraction generated by an initial nucleus, by an originary form that remains unchanged during the process. This circumstance made it possible to connect phýsis to mathematics. This also gave rise to a technique for magnification (or reduction) that would last for centuries – a technique that is recognizable even in the most advanced algebraic computatio.
在希腊，放大几何图形的问题出现在各种轶事中。从埃拉托色尼写给托勒密国王的信中，我们了解到克里特岛国王米诺斯需要将一个立方体形式的皇家陵墓的大小增加一倍，并且在德洛斯建造一个比现有的两倍大的立方体祭坛被阿波罗的神谕规定为抵御瘟疫的一种手段。即使是著名的罗德岛巨像也必须扩大一倍。在希腊，关于放大几何图形的问题没有留存下来的论文，但在希腊传统提供的轶事证据之外，可能还添加了一个并非偶然的因素，它使关于数学起源的猜想更加清晰：印度吠陀仪式思想中数学与宗教的系统结合。Śulvasūtra让我们能够解读在希腊发生的事情的含义。
In Greece the problem of enlarging geometrical figures crops up in a variety of anecdotes. From Eratosthenes’ letter to King Ptolemy we learn that Minos, the king of Crete, needed to double the size of a royal tomb in the form of a cube, and that the construction at Delos of a cubic altar twice the size of the existing one was prescribed by the oracle of Apollo as a means of warding off a plague. Even the celebrated Colossus of Rhodes had to be doubled in size. In Greece there is no surviving treatise on the problem of enlarging geometrical figures, but to the anecdotal evidence afforded by the Greek tradition may be added a factor that is hardly accidental and which allows a conjecture about the origins of mathematics to emerge more clearly: the systematic combination of mathematics and religion in the ritual thought of Vedic India. The theme of scaling up altars in the Śulvasūtra allows us to decipher the meaning of what happened in Greece.
埃拉托色尼解释说，立方体复制的意义在于形式增长的想法。这是一个从最初的核心发展出越来越大的数字的无限发展的问题。成长总是意味着向其他事物转变，这是一种异化和置换的运动，但也可以在相似人物的序列中实现。实际上，几何图形的放大技术允许我们从三角形或正方形构造一个更大的图形，仍然是三角形或正方形。我们将能够以最一般的方式解释 Eratosthenes：
The significance of the duplication of the cube, Eratosthenes explained, was to be found in the idea of the growth of forms. It was a question of developing from an initial nucleus an indefinite progression of figures on an ever-larger scale. Growth always implies change into something else, a movement that alienates and displaces but one that can also be realized in sequences of similar figures. The technique of enlargement of a geometrical figure in effect allows us to construct, from a triangle or a square, a larger figure that is still a triangle or a square. We will be able, in the most general way, explains Eratosthenes:
将每个由平行四边形分隔的给定实体形状变成一个立方体，或者让它从一种形式传递到另一种形式 [ ex hetérou eis héteron ] 并使其相似 [to a given figure] 并在尊重相似性的同时放大它 [ epaúxein ] ，并且在我们处理祭坛和寺庙时也这样做……但是那些希望放大 [ epaúxein ] 弹射器和其他弹道武器的尺寸的人也会发现我的发明很有用，其中一切都必须增加 [ auxethênai ]按比例、宽度、长度……如果我们希望火力和弹道按比例增加。8
to change into a cube every given solid shape that is delimited by parallelograms, or alternatively make it pass from one form to another [ex hetérou eis héteron ] and make it similar [to a given figure] and magnify it [epaúxein ] while respecting similitude, and to do this when we are dealing with altars and temples as well … But my invention will also be found useful by those who wish to magnify [epaúxein ] the dimensions of catapults and other ballistic weapons, in which everything must increase [auxethênai ] in proportion, width, length … if we desire the firepower and trajectory to increase proportionately.8
对于柏拉图来说，数学首先有助于“促进灵魂从生成的世界到真理和存在的世界的彻底转变”（Republic，525 c），但他也声称在数字进展中你可以找到phýsis的印记 - 自然。两者都典型地由流动、形式的渐进起源组成，从能够无限制地扩展和繁殖自身的“种子”开始。正如柏拉图的《美诺》中合理地提出的那样，将正方形的大小加倍的问题具有以下意义：不仅可以通过记忆来举例说明学习的过程，而且还可以在理想的数字中重现phýsis在潜在的无限增长中的力量的类似形式。
For Plato, mathematics served above all to ‘facilitate the radical conversion of the soul from the world of becoming to that of truth and being’ (Republic, 525 c), but he also claimed that in numerical progression you could find the stamp of phýsis – of nature. Both typically consisted of a flow, of a progressive genesis of forms beginning from a ‘seed’ capable of expanding and reproducing itself without limit. The problem of doubling the size of a square, as plausibly proposed in Plato’s Meno, had the following significance: not only to exemplify a process of learning by way of anámnesis but also to reproduce in ideal figures the force of phýsis in a potentially infinite growth of similar forms.
我们需要将几何点的柏拉图概念与phýsis联系起来，正如亚里士多德所指出的（形而上学，992 a），几何点被理解为线的起点和生成原理。柏拉图不接受将点作为静态最小值的想法，作为一个组成部分行或卷。A. E. Taylor 正确地认为这是对后来在希腊柏拉图主义中出现的概念的预想——线被理解为点的流动（rhýsis ），牛顿将这一术语引入英语以表达中心思想17 世纪发展起来的微积分。
We need to connect to phýsis the Platonic conception of the geometrical point, which was understood, as Aristotle points out (Metaphysics, 992 a), as the starting point and the generating principle of the line. Plato did not accept the idea of a point as a static minimum, as a constituent part of a line or a volume. A. E. Taylor rightly sees in this the adumbration of a concept that was to emerge later, in Greek Platonism – of the line understood as the fluxion (rhýsis ) of a point, a term which Newton would introduce to the English language to express the central idea of the calculus that was being developed in the seventeenth century.
牛顿认为数学是一种方法，而不是对事物秩序的有效解释，因此避免了任何试图解释瞬时运动性质的定义。尽管如此，他并没有回避这个概念，而是将其作为他的Methodus Fluxionum et serierum infinitarum或The Method of Fluxions and Infinite Series的基础，该方法写于 1671 年，并于 1736 年首次出版。9在这篇论文中，牛顿将变量描述为无穷小元素的集合体，但作为由点、线和表面的连续运动产生的实体——并称为“流动”，即产生的瞬时速度。在最简单的情况下，术语通量表示沿直线移动的点的瞬时速度。随后，在 1690 年之后，牛顿试图对通量理论给予充分的严谨性，用解析几何来定义其主要概念，与其他学科相比，他所倡导的至高无上。然后，他提出了两个定义，这些定义有助于更清楚地表明增长和缩减思想对西方科学思想的一般历史的重要性：
Newton considered mathematics to be a method rather than an effective explanation of the order of things, and hence avoided any definition which sought to explain the nature of instantaneous motion. Nevertheless, he did not sidestep this concept, making it the foundation of his Methodus fluxionum et serierum infinitarum, or The Method of Fluxions and Infinite Series, written in 1671 and first published in 1736.9 In this treatise Newton described the variable quantities not as aggregates of infinitesimal elements, but as entities generated by the continuous motion of points, lines and surfaces – and called ‘fluxion’ the instantaneous speed of generation. In the simplest case the term fluxion denoted the instantaneous speed of a point that moves along a line. Subsequently, after 1690, Newton attempted to give full rigour to the theory of fluxions, defining its principal concepts in terms of that analytic geometry the supremacy of which he championed in comparison to other disciplines. He then proposed two definitions that help to show even more clearly the importance of the ideas of growth and diminution for the general history of scientific thought in the West:
- Fluens est quod continua mutatione augetur vel diminuitur。
（流利是通过不断变化而增加或减少的东西。）
(A fluent is what is increased or diminished by continuous change.)
- Fluxio est celeritas mutationis illius。
（变化的速度很快。）10
(In fluxion is the swiftness of that change.)10
因此， fluens是通过连续运动增加或减少的速度， fluensio是该运动的速度。对牛顿来说，分析的基本概念起源于与它们出现的瞬间有关的增长和缩小的概念。直到后来，在 19 世纪，微积分基本思想的完全不同的概念才被强加于人，赋予它的不是动态的，而是静态的和原子的特征。由于严格的极限概念，十七世纪微积分形式所代表的势能和动态无穷大被取代，通过一个实际的无限（一个由无限数量的成员组成的“完整”系列），人们希望它更有可能为连续体提供真实的存在。
Hence fluens was that which increases or decreases by means of a continuous movement, and fluxio was the speed of that movement. The fundamental concepts of analysis, for Newton, had their origin in the ideas of growth and diminution related to the instant of their emergence. Only later, in the nineteenth century, did the completely different conception of the fundamental ideas of calculus impose itself, giving it not a dynamic but a static and atomistic character. The potential and dynamic infinity represented by the formalisms of differential calculus of the seventeenth century was replaced, thanks to a rigorous notion of limit, by an actual infinity (a ‘completed’ series consisting of an infinite number of members) that was more likely, it was hoped, to lend a real existence to the continuum.
对柏拉图来说，就像对他之前的毕达哥拉斯学派一样，数字使意识的一种统一活动成为可能，这种活动与感知的多样性有关。在Theaetetus (184 d) 中论证说，特定知觉的多样性以一种混乱的方式存在于我们内部，没有“趋向于一个单一的确定形式，无论是灵魂的形式还是人们可能希望称之为的任何其他形式。 ……我们感知到什么是可感知的”。
For Plato, just as for the Pythagoreans before him, numbers made possible a kind of unifying activity of consciousness with regard to the multiplicity of perceptions. It is argued in the Theaetetus (184 d) that the multiplicity of particular perceptions resides within us in a confused way, without converging ‘towards a single determined form, whether that of a soul or whatever else one might wish to call it, with which … we perceive what is perceptible’.
灵魂如何运作？柏拉图在《泰阿泰德》（186 a-b）中说，存在于所有事物中的存在正是灵魂本身所倾向于的存在，在对立的事物之间建立关系（类比 ）并将过去和现在与自身并列未来。因此，灵魂的功能之一在于在两种不同的知觉之间建立联系，这些知觉在时间上一个接一个地出现，而理想的联系就是引导我们在同一事物转变为另一事物时识别出同一事物的联系。只有那些东西是可以理解的尽管可能会发生变化，但我们仍然可以识别。在没有允许识别可变实体的数字法则的情况下，感知流会退化为无法弥补的混乱。11
How does the soul operate? The being that is present in all things, says Plato in the Theaetetus (186 a–b), is precisely that towards which the soul itself inclines, establishing relations (analogízein ) between antithetical things and juxtaposing in itself the past and the present with the future. Hence one of the functions of the soul lies in establishing the nexus between two distinct perceptions that come one after the other in time, and the ideal connection is then that which leads us to recognize the same thing in its transformation into something other. Only those things are intelligible which, although subject to change, remain recognizable to us. In the absence of a numerical law that allows the recognition of changeable entities, the flow of perceptions degenerates into irremediable chaos.11
现在，借助数学，可以考虑在不同感知之间的稳定联系中捕捉存在。在 20 世纪的数学家中，Brouwer 完美地理解，思想从中呈现其形式的第一个现象是一个时间序列，在该序列中，我们的意识在移动到下一个感觉的行为中保留了主要感觉。对于 Brouwer 来说，这种现象是递归地重复的，我们的意识将过去的感觉与现在的感觉区分开来，并从两者中退后一步，从而引发了我们的思想活动。这个过程的数学形式可以精确地在迭代计算的过程中找到，其中变量以一系列近似值逼近问题的解，每一个近似值都是通过重复应用算子从前一个值计算出来的. 顺便说一句，Norbert Wiener 在模拟生物体的行为时想到了类似的迭代机制，特别是在发生反馈现象的情况下。他指出，对物体的抓握在于手和物体之间的剩余距离的逐渐减小，正如在迭代过程中，随着一系列操作的重复，变量值之间的剩余距离并且寻求的解决方案逐渐减少。
Now, with mathematics, it is possible to think of capturing existence in a stable link between distinct perceptions. Among twentieth-century mathematicians, Brouwer understood perfectly that the first phenomenon from which thought assumes its form is a temporal sequence in which our consciousness retains a primary sensation in the act of moving to a subsequent one. For Brouwer the phenomenon is repeated recursively, and our consciousness, distinguishing the past sensation from the present and standing back from both, gives rise to the activity of our mind. A mathematical form of this process can be found precisely in the procedures of iterative calculation, in which the variable approaches the solution of a problem with a succession of approximate values, each of which is calculated from the previous one by the repeated application of an operator. Incidentally, Norbert Wiener thought of analogous iterative mechanisms in the simulation of the behaviour of living organisms, especially in cases in which the phenomenon of feedback occurred. The grasping of an object, he noted, consists in the progressive reduction of the residual distance between hand and object, exactly in the way that in iterative processes, with their repetition of a series of operations, the residual distance between the value of the variable and the solution sought is gradually reduced by degrees.
5. Katà gnómonos phýsin : Gnomon 的本质
5. Katà gnómonos phýsin : The Nature of the Gnomon
那么我们应该如何思考数字，以完成将灵魂的感知连接在一起的任务呢？数字可以用多种方式来思考，但前苏格拉底的菲洛劳斯有一个片段向我们展示了正确的方式——最合适的方式，也就是我们聚合思想的能力，以及从中获得数学的迭代算法起源。根据这个片段，数字“将所有事物与灵魂的内在感知相协调，使它们在彼此之间可识别和可比，根据晷针的性质 [ katà gnómonos phýsin] ]，因为它形成和分解了事物之间的所有个体关系，那些不受限制的事物与那些受限制的事物一样多”（44 B 11 DK）。在这里，我们找到了所有决定性的元素：数字、灵魂、事物 ( pragmata )、知觉、可知性、关系、有限的概念、无限的概念，最后是晷针的本质。指针，亚历山大的苍鹭将继续解释（定义，58），是一个图形，当应用于给定的几何形状时，会生成与第一个形状相似的另一个形状。毕达哥拉斯数在空间中按一定的几何顺序排列，它们的生长符合晷针的性质：数字 4 由四个排列成正方形的点组成；用沿其两侧排列的五个点围绕它们，一个得到随后的平方数，即9。同理，一个得到 16, 25, 36, 49 ... 每次加上奇数个点排列成一个正方形。感谢 George Thibaut 在 1870 年代 Baudhāyana 的Śulvasūtra的翻译，我们可以完全理解吠陀祭坛的砖块和毕达哥拉斯数字之间的类比。建造烈火祭坛的说明包括以下内容：从一个由四块砖组成的小方块开始；然后，通过添加五块砖，继续组成九个正方形；再一次，加上七块砖，跟着它是一个十六的正方形。1确定祭坛尺寸的人显然设想了平方数可能无限增长。
So how should we think about numbers, in order to accomplish the task of linking together the soul’s perceptions? Numbers can be thought about in a variety of ways, but there is a fragment by the pre-Socratic Philolaus that shows us the right way – the most appropriate, that is, to our capacity for aggregating our thought, and the one from which the iterative algorithms of mathematics originate. Numbers, according to this fragment, ‘harmonizing all things with the internal perception of the soul, makes them recognizable and commensurable among themselves, according to the nature of the gnomon [katà gnómonos phýsin ], because it forms and disassembles all the individual relations between things, those that are without limit as much as those that are limited’ (44 B 11 DK). Here we find all the decisive elements: numbers, the soul, things ( prágmata ), perception, knowability, relations, the notion of the finite, the infinite and, finally, the nature of the gnomon. The gnomon, Heron of Alexandria would go on to explain (Definitions, 58), is the figure which, when applied to a given geometrical shape, generates another shape similar to that first shape. Pythagorean numbers were arranged in a certain geometrical order in space, and their growth was in keeping with the nature of the gnomon: the number 4 consisted of four points arranged in a square; surrounding them with five points arranged along two of its sides, one obtains the subsequent square number, that is to say, 9. In the same way one obtains 16, 25, 36, 49 … adding each time an odd number of points arranged to form a square. Thanks to George Thibaut’s translation in the 1870s of Baudhāyana’s Śulvasūtra, we can perfectly understand the analogy between the bricks of the Vedic altars and Pythagorean numbers. The instructions for building the altar of Agni consisted of the following: start with a small square made up of four bricks; then, by adding five bricks, go on to make a square of nine; then again, with the addition of seven bricks, follow it with a square of sixteen.1 Whoever established the measurements of the altar evidently conceived of the potentially infinite growth of square numbers.
为什么这种排列数字的方式能够实现预先确定的让事物可知的目的？为了为了理解，有必要在几何连续体中再现通过日光扩大的想法，特别是考虑正方形。欧几里得使用这种放大的图像来演示二项式平方的代数公式的几何版本：将一条直线分成两段a和h，整条直线a + h上的平方相等到a的平方加上h的平方，再加上边a和 h的矩形的两倍，一个定理，由 Āpastamba 的Śulvasútra 中提出的简单的正方形扩大公式表示： ( a + h ) 2 = a 2 + 2 ah + h 2。Newton 和 Raphson 将继续使用这个公式在数值上求解一个二次方程，例如x 2 − 2 = 0。给定/假设 x = a + h，其中a是
Why does this way of arranging numbers fulfil the pre-established purpose of making things knowable? In order to understand, it is necessary to reproduce in the geometrical continuum the idea of enlargement through gnomons, thinking in particular of the square. The image of just such an enlargement was used by Euclid to demonstrate the geometric version of the algebraic formula for the square of a binomial: dividing a straight line into two segments, a and h, the square on the whole line a + h is equal to the square of a plus the square of h, plus two times the rectangle of the sides a and h, a theorem that is expressed by the simple formula of enlargement of the square proposed in Āpastamba’s Śulvasútra : (a + h )2 = a2 + 2ah + h2. Newton and Raphson would go on to use this formula to resolve numerically a quadratic equation such as x 2 − 2 = 0. Given/supposing that x = a + h, where a is an initial approximation of 2, and disregarding h2, the equation becomes a2 + 2ah = 2, from which it follows that h=−a2−22a. And from this we get the iterative method xk+1=xk−(xk2−2)2xk where k = 0, 1, 2 … and x0 = a.
图 2
Figure 2
欧几里得定理（Elements , II, 4）暗示的远比它实际陈述的要多得多，我们可能想知道欧几里得本人实际上意识到这一点有多远。Philolaus 的katà gnómonos phýsin表达片段中的含义需要与phýsis的基本含义联系起来，即涉及增长，实际上，欧几里德告诉我们，当边a与线段h相加时，正方形如何增长。以这种方式解释，定理的含义发生了根本性的改变。段a和h的含义不同，因为h被解释为a的增量. 分析从这里开始：一般来说，如果x经历增量变化h，我们会问函数f ( x ) 的值如何变化。Euclid (II, 4) 向我们展示了简单函数f ( x ) = x 2的情况，我们问自己x的平方如何随x的变化而变化。但是这个问题在范围上是完全一般的：它是关于一个依赖于另一个量B的量A发生正或负增加时如何发生增长（或减少）。这个问题的第一个全面答案可以在著名的论文Methodus incrementorum directa et inversa（“增量的直接和间接方法”，1715 年在伦敦出版）中找到，其中布鲁克泰勒介绍了一个公式，该公式表示函数f ( x ) 随变量x的小增量而变化. 拉格朗日后来将泰勒公式定义为微积分的基本原理。在一个方向或另一个方向上，x的增量可以确定函数f的减小，从而我们获得了一类求x值的方法，通过该方法f取其最小值。我们可以从莱昂哈德·欧拉 (Leonhard Euler) 的名言中衡量这一点的重要性，根据他的说法，宇宙的完美体现在这样一个事实，即世界上没有任何事情会发生不涉及最小或最大规则的事情。
Euclid’s theorem (Elements, II, 4) implies a great deal more than it actually states, and we may wonder how far Euclid himself was actually aware of this. The sense in Philolaus’ fragment of the expression katà gnómonos phýsin needs to be connected with the fundamental meaning of phýsis as involving growth, and indeed Euclid tells us how a square grows when a side a is added to by a segment h. Interpreted in this way, the meaning of the theorem is radically altered. The segments a and h do not have the same meaning, because h is interpreted as an increment of a. From here the analysis begins: in general, we ask how the value of a function f (x ) changes if x undergoes the incremental change h. Euclid (II, 4) shows us the case of the simple function f (x ) = x 2, and we ask ourselves how the square of x varies with the variation of x. But the problem is altogether general in scope: it is about how a growth (or a diminution) occurs to a quantity A that depends on another quantity B when the latter undergoes an increment, positive or negative. The first comprehensive answer to the problem is to be found in the celebrated treatise Methodus incrementorum directa et inversa (‘Direct and Indirect Methods of Incrementation’, published in London in 1715), in which Brook Taylor introduced a formula which expresses how a function f (x ) varies for small increments of the variable x. Lagrange would later define Taylor’s formula as the fundamental principle of differential calculus. In one direction or another, the increments of x can determine the decrease of the function f, and we thereby obtain a class of methods for finding the value of x by which f assumes its minimum value. We can gauge the importance of this from the celebrated words of Leonhard Euler, according to whom the perfection of the universe was expressed in the fact that nothing happens in the world without some rule of minimum or maximum being involved.
很难想象，在没有这种分析设备的情况下，如何测量一个量在改变另一个量后所受到的扰动的影响。在分析公式中，我们发现了用于研究面临数据和单个操作中的破坏性错误的计算系统稳定性的第一个工具。二项式平方的公式可以考虑从a到a + h的两个连续变化时刻，并从初始平方重新发现另一个平方。这意味着一种稳定性，一种认识论所基于的变化的不变性，以及思考某事的心灵的力量。改变不仅是在（到）别的东西（eisállo ）；成为不仅仅是非存在。在这种观点中，phýsis变成了类似形式的迭代生成流程，或者至少我们相信这可能是这样的——并且研究了确实是这种情况的例子。但是在这个关头，代数微积分已经宣布了自己，它以几何图形的增长为模型，其中phýsis的增长和生成现象可能已经找到了数学表示。同样的规则也适用于我们的日常经验：没有重复，正如普鲁斯特所说，不真实占了上风。2
It is difficult to imagine how it would be possible, in the absence of such an analytical device, to measure the effect of the disturbance that a quantity undergoes following the alteration of another quantity. In the analytic formulas we find the first instruments for studying the stability of a system of calculation faced with disruptive errors in the data and in individual operations. The formula for the square of the binomial makes it possible to consider two successive moments of change, from a to a + h, and to rediscover another square from an initial one. This implies a stability, an invariance in change on which the epistéme is based, the power of the mind to dwell on something. Change is not only in(to) something else (eis állo ); becoming is not just non-being. In this view, phýsis becomes a flow of iterative generation of similar forms, or at the very least we are convinced that this might be so – and examples are studied in which it is indeed the case. But already at this juncture an algebraic calculus announces itself, modelled on the growth of geometrical figures, in which the phenomena of the growth and generation of phýsis might have found a mathematical representation. And something like the same rule applies to our everyday experience: without repetition, as Proust has it, unreality prevails.2
从二项式平方的公式外推以数值求解代数方程，Viète、Newton 和 Raphson 开发的方法得到了公正的赞誉。这是求解方程和计算函数最小值的最强大和最有效的机器——在现代，数学发现自己根据古代制定的计算法则重新阐述了这一点。就在几十年前，从收敛速度和计算复杂度的角度来看，牛顿近似数平方根的方法被证明是最有效的。同样最近的是计算泛化牛顿模式的函数最小值的方法的发展。3
Extrapolated from the formula for the square of the binomial in order to solve an algebraic equation numerically, the methods developed by Viète, Newton and Raphson are justly celebrated. This was the most powerful and efficient machine for solving equations and calculating the minimum value of a function – something mathematics found itself re-elaborating, in modern times, following computational laws laid down in antiquity. Just a few decades ago it was demonstrated that Newton’s method for approximating the square root of a number is the most efficient possible, measured in terms of speed of convergence and computational complexity. Equally recent is the development of methods of calculation of the minimum value of a function which generalizes Newton’s schema.3
然而，在这一点上，我们不能忽视一个关键的特征，它表现在数字计算和许多其他形式中，是数值算法不稳定的主要原因之一。在由牛顿近似方程根的方法生成的分数p / q中，分子p和分母q会迅速增长到超出可容忍的范围，从而导致计算成本增加，并且由于以下原因导致信息丢失四舍五入的过程。4在平方根或立方根的计算中，数字的增长与几何形状通过连续的指法校正而增长或缩小有关，并且在该过程的原始逻辑中，与测量单位的逐渐缩小有关。
At this point, however, we should not neglect to mention a crucial characteristic, which is manifested in digital calculation and in many other forms, and is one of the main causes of the instability of numerical algorithms. In the fractions p/q generated by Newton’s method for approximating the root of an equation, the numerator p and the denominator q can rapidly grow beyond tolerable limits with a consequent increase in the computational cost, and with a loss of information due to the process of rounding off.4 In the calculation of square or cubic roots the growth of numbers is linked to the growth or diminution of a geometrical shape through successive gnomonic corrections, and, in the original logic of the process, to a progressive diminishing of the unit of measurement.
今天，用于计算函数最小值和最大值的数值优化策略基于 Viète 和 Newton 使用的相同方法方案。并且基于这些相同的策略依赖于神经网络的自动学习过程，该过程基于参数的计算，以逐渐减少计算结果和预期答案之间的差距。从这个意义上说，学习，或者更好的，由神经网络实现的学习模型，在概念上与最小化过程相同。但解析计算并不总能保证手术成功。在学习的自动过程中，由于矩阵的病态，可能经常发生数值的不受控制的增长，从而导致计算结果的致命意义丧失。
Today the strategies of numerical optimization for calculating minimum and maximum values of functions are based on the same scheme of methods used by Viète and by Newton. And on these same strategies depend the processes of automatic learning with neural networks, which are based on the calculation of the parameters for progressively reducing the gap between calculated and expected answers. In this sense, learning, or better, the model of learning realized by the neural network, is conceptually identical to a process of minimization. But analytical calculation is not always able to guarantee the success of the operation. In the automatic processes of learning, uncontrolled growths of numerical values may often occur, due to the ill-conditioning of matrices, with a consequently fatal loss of meaning for the results of the calculation.
6. Dýnamis：生产能力
6. Dýnamis: The Capacity to Produce
在古希腊，通过扩大正方形和其他几何图形，表达了一种积极的生产，几乎是一种萌芽，就像奥菲斯的phýsis所暗示的那样。在dýnamis （潜力）中发现了一个类似的想法，通过它，表面延伸到线之外。Euclid 展示了如何在一条直线上构造一个正方形，即从一侧提取图形。1 Proclus（对欧几里得的“元素”的第一本书的评论，423-4）后来将评论构造（systésasthai ）三角形和通过几乎产生一个正方形（ anagráphein apó ）之间的区别它从一个侧面。在美索不达米亚数学中，出现了一个含义不确定的术语 ——takīltu—— 根据 Thureau-Dangin 的假设，它的意思很可能是“产生某物的东西”，就像一个数字乘以自身产生一个平方。2毕达哥拉斯定理，其证明可在欧几里得元素(I, 47) 中找到，很可能是对保持其形式不变的几何图形增长标准的回应。从两个正方形我们构建一个更大的第三个，前两个的总和，从矩形的对角线生成它，矩形的两侧与两个正方形的边重合。增长和生产似乎是由同一条规律统一起来的。
In ancient Greece, through the enlargement of a square and other geometrical figures, an active production was expressed, almost a germination, such as the one to which the phýsis of the Orphics alludes. An analogous idea is found in the dýnamis ( potentiality) by which a surface extends outside a line. Euclid showed how to construct a square on a straight line, in the sense of extracting a figure from one side.1 Proclus (Commentary on the First Book of Euclid’s ‘Elements’, 423–4) would later remark on the difference between constructing (systésasthai ) a triangle and tracing (anagráphein apó ) a square by almost producing it from a single side. In Mesopotamian mathematics, a term of uncertain meaning – takīltu – occurs that could well mean, according to Thureau-Dangin’s hypothesis, ‘that from which something has been produced’, like a number that when multiplied by itself produces a square.2 Pythagoras’ theorem, the demonstration of which is to be found in Euclid’s Elements (I, 47), could well be a response to the criterion of the growth of geometric figures that keep their form unaltered. From two squares we build a larger third, the sum of the first two, producing it from the diagonal of the rectangle the two sides of which coincide with those of the two squares. Growth and production seem to be united by the same law.
毕达哥拉斯定理在古代美索不达米亚数学中广为人知，尽管没有像欧几里得那样证明它的记录。巴比伦人试图解决，通过算法，简单的具体问题不具备我们在欧几里得定理中遇到的抽象程度和普遍性。相反，巴比伦的计算显示出一种计算模式的命令，这些模式既有效又持久，今天仍然是我们用来解决复杂数学问题的最先进计算的基础。巴比伦算术的一个典型问题是计算已知高度和宽度的矩形门的对角线。抄写员将解决方案与扩大正方形的问题联系起来，这提出了一种类似于现代数学求解代数方程所采用的计算方案。3因此，为了遵守计算标准，毕达哥拉斯定理明确地与几何图形的增长相关联。
Pythagoras’ theorem was known in ancient Mesopotamian mathematics, although there is no record of a demonstration of it such as Euclid’s. The Babylonians tried to solve, by means of algorithms, simple concrete problems that did not possess the degree of abstraction and generality that we encounter in Euclid’s theorems. Instead the Babylonian calculations show a command of computational schemas that are as efficient as they are enduring, and which are still today the basis of the most advanced calculations we use to solve complex mathematical problems. A problem typical of Babylonian arithmetic consists of calculating the diagonal of a rectangular door of which the height and width are known. The scribe connects the solution back to the problem of the enlargement of a square, which suggests a scheme of calculation analogous to that which modern mathematics would adopt in order to solve an algebraic equation.3 Consequently, in order to observe the criteria of calculation, Pythagoras’ theorem is explicitly connected to the growth of a geometric figure.
在 Āpastamba 的Śulvasūtra (I, 4-5) 中，毕达哥拉斯的相同定理被表述为表明矩形的对角线在其自身之外产生（ karoti ）较小边和较大边各自产生（ kurutas ）外部。4
In Āpastamba’s Śulvasūtra (I, 4–5) the same theorem of Pythagoras is formulated in such a way as to show that the diagonal of a rectangle produces (karoti ) outside itself that which the smaller side and the larger side each produce (kurutas ) externally.4
在希腊语poieîn （生产）中，我们掌握了建筑的基本原理，类似于梵语vi-hṛ 5增加尺寸的几何图形，根据古代几何学以及古代和现代数值计算的标准时代。在poieîn中存在高效操作的概念：例如，我们可以有效地计算乘法的表达式，或者我们今天所说的——并非偶然——“乘积”。
In the Greek term poieîn (to produce) we catch the elementary principle of building, analogous to the Sanskrit vi-hṛ 5 geometrical figures of increasing dimensions, according to criteria that have marked ancient geometry as well as numerical calculation in both antiquity and the modern era. In poieîn there is the idea of operating efficiently: we can for instance calculate in an effective way the expression of a multiplication, or of what we call today – not accidentally – the ‘product’.
在希腊数学中，术语dýnamis指的是成为或成为某物的能力，表示正方形的边 - 也就是说，测量面积的数字的平方根。因此，由dýnamis表示的线也可能表示它们构成边的正方形：3 的平方根可以表示面积为 3 的正方形（Theaetetus，147 d ff.）。
In Greek mathematics the term dýnamis, which refers to the capacity to be or to become something, denotes the side of a square – that is to say, the square root of the number that measures the area. Hence the lines connoted by a dýnamis may also mean the squares for which they compose a side: the square root of 3 can denote the square with an area of 3 (Theaetetus, 147 d ff.).
图 3
Figure 3
在《政治家》 （266 a-b）中，柏拉图提到了对角线，以及对角线的对角线，仿佛这是一个无限渐进生成的问题，即初始边有能力生成的正方形的倍增。
In the Statesman (266 a–b) Plato mentions the diagonal, and the diagonal of the diagonal, as if it was a question of indefinite progressive generation, of the doubling of a square that the initial side has the power to generate.
在俄耳甫斯和毕达哥拉斯的传统（Philolaus，44 B 11 DK 和 315 Kern）中，这种权力观念也包含在十进制的完美之中，这是“神圣、天体和人类生命的原则”，没有它“一切事物都将是无限的、模糊的和不确定的”。
In the Orphic and Pythagorean traditions (Philolaus, 44 B 11 DK and 315 Kern) this idea of power was also contained within that of the perfection of the decad, which was the ‘principle of divine, celestial and human life’, without which ‘all things would be limitless, obscure and uncertain’.
与巴比伦古代数学中的dýnamis相对应的概念是mithartum，即由其平方根 ( íbsi ) 产生的几何正方形。6在几何中吠陀祭坛 连续正方形的生成，一个来自另一个的对角线，由术语dvikaranī 表示，对角线产生双，7或者，根据柏拉图的美诺的指示，一个不确定的正方形连续，一个之后另一个，也就是说，一个的对角线等于下一个的边。在所有这些情况下，产生现象都是必不可少的：一个数学实体产生另一个，好像所需要的是这些实体本身固有的逐步扩大的力量。类似地，在更晚的日子里，有人声称分数的算术产生了允许我们识别和定义新的数字实体的现象：由 Dedekind 定义为部分的实数将仅由有理数的性质产生。
The notion corresponding to dýnamis in the mathematics of Babylonian antiquity was mithartum, the geometric square produced by its square root (íbsi ).6 In the geometry of the Vedic altars the generation of successive squares, one from the diagonal of the other, is rendered by the term dvikaranī, the diagonal which produces the double,7 or, according to the indication of Plato’s Meno, an indefinite succession of squares, one after another, that is to say, with the diagonal of one equal to the side of the next. In all of these cases the phenomenon of production is essential: a mathematical entity produces another, as if what was entailed consisted of a power of progressive enlargement inherent in those entities themselves. Similarly, at a much later date it will be claimed that the arithmetic of fractions produces phenomena that allow us to recognize and to define new numerical entities: real numbers, defined by Dedekind as sections, will be produced by properties of only rational numbers.
海德格尔抓住了poiésis的意义，即生产的意义，作为希腊技术的中心节点，并指出在这个语境中，每一个生产都是基于“揭开”的真理。根据海德格尔的说法，现代“技术”的情况完全不同，在这种情况下，揭幕不是一种生产（ poiésis ），而是一种挑衅（Herausfordern ），它不可避免地导致试图利用工业规模的能源，根据最大效用和最小成本的退化标准。8允许这种关键转变，这种生产过程退化的算法，仍然依赖于古代数学的相同基本运算，精确地委托给生产。这些操作在古代知识中所具有的意义的复兴导致提取，可以说，它们最枯燥的方面：对技术功能的目标最有帮助的方面。
Heidegger grasped the meaning of poiésis, of production, as the central nodal point of Greek téchne, specifying that every production is based in this context on the truth of ‘unveiling’. Quite different, according to Heidegger, is the case of modern ‘technique’, in which the unveiling is not a production ( poiésis ) but a provocation (Herausfordern ), which results inevitably in the attempt to exploit energy on an industrial scale, according to the degenerate criterion of maximum utility with minimum cost.8 The algorithms that allow this critical transition, this degeneration of the process of production, still rely upon the same elementary operations of ancient mathematics, the ones delegated, precisely, to production. The revitalization of the meaning that these operations possessed in the knowledge of antiquity resulted in the extraction, so to speak, of their most arid aspect: the one most instrumental for the aims of technological functionality.
图 4
Figure 4
数学动态的本体论含义从柏拉图的《智者》 （247 d-e）中的一段话中清楚地体现出来：
The ontological implications of mathematical dýnamis emerge clearly from a passage in Plato’s Sophist (247 d–e):
因此，我断言，就其本质而言，任何事物都具有（dýnamin ）产生（ poieîn ）任何效果或受制于它的能力，即使是最微不足道的事物和最小的程度，即使只有一次，这一切都存在于现实（toûto óntos eînai ）。我实际上提出了一个定义：实体只不过是权力（dýnamis ）。
I therefore assert that everything which by its nature possesses a capacity (dýnamin ) to produce ( poieîn ) any effect or to be subject to it, even the most insignificant thing and to the smallest degree, and even if only once, all this exists in reality (toûto óntos eînai ). I propose in fact a definition: entities are nothing other than power (dýnamis ).
Sophist中的 Eleatic Stranger 说，存在的过程恰恰需要一种生产行为，而产生的东西必须被定义为“产品”（Sophist，219 b）。共和国中的苏格拉底(477 c) 进一步指出，在dýnamis我们可以找到事物的类型 ( génos ti tôn ónton )，Proclus 逐字重复了这一观察结果 ( Commentary on the 'Republic of Plato , I, 266, 17) ）。The Eleatic Stranger of the Sophist (238 a) 将“整个数字领域置于存在的事物之中”，以及dýnamis的含义作为一种扩展的力量和作为自然生成的原则，作为一个phýsis的移动媒介，它在模型上真正地发芽实体数字级数的概念，在柏拉图的Epinomis (990 c)中得到了明确的说明：
The course of being, says the Eleatic Stranger in the Sophist, entails precisely an act of production, and that which is brought into being must be defined as a ‘product’ (Sophist, 219 b). Socrates in the Republic (477 c) further noted that in dýnamis we can find the type of things ( génos ti tôn ónton ), an observation that was repeated verbatim by Proclus (Commentary on the ‘Republic’ of Plato, I, 266, 17). The Eleatic Stranger of the Sophist (238 a) placed ‘the entire province of numbers among the things that have being’, and the meaning of dýnamis as a power of extension and as a principle of natural generation, as the moving agent of a phýsis which literally germinates entities on the model of numerical progression, is made explicit in Plato’s Epinomis (990 c):
现在主要和最重要的研究是数字本身，而不是有形的数字，而是每个可能的世代和增长潜力（dynameos ）[从正方形和立方体的高度]，奇数和偶数，以及数字对事物本质（ phýsis ）的影响。
Now the primary and most important study is of numbers in themselves, not of corporeal numbers, but of every possible generation and potentiality (dynámeos ) of growth [with elevations to the square and to the cube], of the odd and the even, and of the influence that numbers have on the nature ( phýsis ) of things.
在Philebus (15 d; 25 d–e) 中，柏拉图通过lógos，即数学关系 ( tautòn hèn kaì pollà hypò lógon ) 看到了一与多巧合的矛盾可能性，并且已经赋予数字和关系以结束对立面之间对立的能力。因此，在数字中，所表达的不是单纯的多样性，而是由于关系性而实现了实际事物中固有的一与多之间的综合。柏拉图宣称，这是“古代人传给我们的知识，他们比我们更好，更接近众神”（Philebus，16 c）。在蒂迈欧(52 c) 此外，他还补充说，是lógos负责以真理和准确的方式告诉我们“它的真实面目”，表明只要一件事是一件事，另一件事是另一件事，那么两者中的任何一个都不能进入另一个以保持其自身并同时成为两个'。因此，逻辑不仅仅是一种话语，而是以更具体的方式，数学被赋予构建适当定义的角色的关系或相互作用。数学关系是将一件事与另一件事联系起来的工具，将一和二放在一起，正如一系列相似的图形所证明的那样，这些图形很多，但由于形式的不变性，它们是相同的。根据亚里士多德的指示（主题，158 b），类似的级数通常可以干预相同的关系定义，如在anatanaíresis的情况下，或在欧几里得算法中用于计算两个量级的比率。柏拉图认为，如果没有数字知识，我们将失去了解真实的可能性，并沦为只有感知和记忆（Epinomis，977 c）。
In the Philebus (15 d; 25 d–e) Plato had seen a paradoxical possibility of coincidence of the one and the many by way of the lógos, that is to say, of mathematical relation (tautòn hèn kaì pollà hypò lógon ), and had ascribed to numbers and relations the capacity to end the oppositions between contraries. In numbers, therefore, what is expressed is not a mere multiplicity but on the contrary the realization, thanks to relationality, of the synthesis between the one and the many that is inherent in actual things. This, Plato declares, is knowledge bequeathed to us by the ‘ancients, who were better than us and lived closer to the gods’ (Philebus, 16 c). In the Timaeus (52 c) he adds, furthermore, that it is the lógos that is responsible for telling us with truth and exactitude ‘that which it really is, showing that, as long as a thing is a thing and another thing is another thing, then neither of the two could enter into the other so as to remain itself and at the same time become two’. Hence the lógos is not simply a discourse, but in a more specific way the relationship or interplay whereby mathematics is given the role of constructing appropriate definitions. The mathematical relation is the instrument which interrelates one thing with another, putting one and two together, exactly as is demonstrated by a progression of similar figures which are many and yet, due to the invariability of the form, one and the same. Similar progressions can typically intervene in the same definition of relation, according to Aristotle’s indications (Topics, 158 b), as in the case of anatanaíresis, or in the Euclidean algorithm for the calculation of the ratio of two magnitudes. Without a knowledge of numbers, Plato believed, we would lose the possibility of knowing the real and be reduced to only perceptions and memories (Epinomis, 977 c).
面对可能被粉碎成流形的可能性，数字是展开存在的最后一道防线。因此，毫不奇怪，在 Archytas 的片段中发现，计算科学在知识方面比其他学科具有明显的优势，并且能够解决它想要的任何事情。一种比几何本身更清晰、更明显的方式（ enargéstero ）。希腊语enargés指的是某种以透明、可触知的方式出现并可以抓握的东西，几乎类似于有形的形式。在特殊情况下，众神本身可以坚持这样的幻影（奥德赛，十六，161）。Iamblichus 将继续观察毕达哥拉斯将数定义为扩展和驱动单元中内在的开创性lógos 。9一个重要的证词，帮助我们理解数字作为phýsis 固有的成长过程的重要性；Orphics 习惯于说这是一个逐渐发生的过程，就像网的啮合（亚里士多德，《动物的世代》，734 a）或一种渐进的分支（316 Kern）。
Against potential pulverization into the manifold, numbers were the last defence of an unfolding existence. It is not surprising, then, to find that in the fragment by Archytas, it is claimed that the science of calculation has, with respect to knowledge, a clear-cut superiority in comparison to other disciplines and is capable of tackling anything it wants in a way that is clearer and more obvious (enargéstero ) than geometry itself. The Greek term enargés refers to something that appears and is graspable in a transparent, palpable way, almost akin to a corporeal form. In exceptional cases the gods themselves could adhere to such apparitions (Odyssey, XVI, 161). Iamblichus would go on to observe that Pythagoras had defined number as the extension and actuation of the seminal lógos immanent in the unit.9 An important testimony that helps us to understand the significance of numbers as a process of growth inherent in phýsis ; a process that takes place gradually, the Orphics were wont to say, like the meshing of a net (Aristotle, On the Generation of Animals, 734 a) or in a sort of progressive ramification (316 Kern).
综上所述，这些线索似乎表明数字被认为是一个物理实体，尽管当然不是因为缺乏抽象能力。相反，数字是资源，是事物的现实性和现实性背后的最终原因，是它们在世界上存在的证据的原则，它反对所有可能的支持现实的虚幻本质的论据。出于这个原因，人与神之间的联系的功能属于数字作为能够代表明显的、无形的真理的命令的实体。10
Taken together, these clues seem to suggest that numbers were conceived of as a physical entity, though certainly not for lack of a capacity for abstraction. On the contrary, number was the res vera, the ultimate reason behind the actuality and reality of things, the principle of the evidence of their existence in the world, against all possible arguments in favour of the illusory nature of the real. For this reason, the function of the link between the human and the divine belonged to numbers as entities capable of representing a command of evident, intangible truth.10
正如西蒙娜·威尔（Simone Weil）所写：“意识和现实与在单一精神运作中同时捕获的系统的多样性成正比”。11现在，一直是数字允许这种操作以多种方式发生，无论几个世纪以来在不同背景下如何理解数字的概念：作为结合同一类不同实体的原则；作为可递归定义的单个函数中各种单独操作的综合；作为一个实数，原子，密集抽象场中的一个元素，有序而完整；作为理性领域的一部分，类似于欧几里德的关系概念；作为一个根据晷针的性质构想的数字，它在几何图形的成长过程中实现了两个不同时刻之间的联系，从而允许在经历了它所经历的变化后识别出相同的图形。最后，
As Simone Weil wrote: ‘consciousness and reality are proportional to the multiplicity of systems captured simultaneously in a single operation of the spirit’.11 Now it has always been numbers that have allowed this operation to take place in a variety of ways, regardless of how the notion of number was understood in different contexts over the course of centuries: as a principle of combining different entities of the same class; as the synthesis of varied, individual operations in a single function definable recursively; as a real number, atomic, an element in a dense abstract field, ordered and complete; as a section of the rational field, analogous to the Euclidean concept of relation; as a number conceived according to the nature of the gnomon, which fulfils a nexus between two distinct moments in the growth of a geometrical figure, thereby allowing the same figure to be recognized after undergoing the changes to which it has been subjected. And finally, there is the notion of number as a computational process, as an algorithm that must respond to precise requirements of efficiency.
普罗提诺断言，所有可理解的事物在某种意义上也是真实的事物（kaì noerà kaí pos tò prâgma ）。所以它适用于表面、实体和所有形状，关于它们的位置和方式 ：
Plotinus asserted that all intelligible things are in a certain sense also real things (kaì noerà kaí pos tò prâgma ). So it was for surfaces, solids and all the shapes, in regard to their where and their how :
事实上，形状不仅由我们构思。它们由出现在我们面前的宇宙的形状来证明，不亚于在自然事物[phýsis]中的其他自然形状[physikàschémata]，它们 必然 存在 于上面[ anánke ]，在身体之前，在它们的纯度，作为主要形式 [ prota schémata ]。( Enneads , VI, 6, 17, 20–25)
Shapes, in fact, are not only conceived by us. They are attested to by the shape of the universe that comes before us, no less than by the other natural shapes [ physikà schémata ] that are in the things of nature [ phýsis ] that exist by necessity [anánke ] above, before bodies, in their purity, as primary forms [ prôta schémata ]. (Enneads, VI, 6, 17, 20–25)
决定性因素，将数学与现实联系在一起的主要原因，是“无限与数相反”（Enneads , VI, 6, 17, 4）（即使我们说数是无限的，因为每个当我们想到一个时，我们也可以想到一个更大的）。数字的概念和数字之间的关系以极限为特征，因此它们与ápeiron相对，是无限的，与缺席 ( stéresis )、不真实和不存在同义。对于普罗提诺来说，无限有两种形式：一种是模型，另一种是图像，而后者最恰当地称为无限（Enneads II, 4, 15, 20-25）。物质 ( hýle ) 是无限的，因为它反对lógos，也就是说，反对关系或联系，在衡量一个事物与另一个事物时，它使现实事物在空间和时间中的有序布置成为可能。现代数的概念就是从关系的概念衍生出来的。
The decisive factor, the principal reason for the bond that united mathematics and reality, was that ‘the infinite is in contrast to number’ (Enneads, VI, 6, 17, 4) (even if we say that number is infinite, because every time we think of one we can also think of a bigger one). The notion of number and the relation between numbers were characterized by a limit, and they were consequently the opposite of the ápeiron, of that which is without limit, synonymous with absence (stéresis ), unreality and non-being. For Plotinus, there were two forms of infinity: one was the model, the other the image, and it was the latter that could most properly be called infinite (Enneads II, 4, 15, 20–25). Matter (hýle ) was infinite because of its opposition to lógos, that is to say, to relation or connection, which in the measuring of one thing with another rendered possible instead an ordered disposition of real things in space and time. It is from the concept of relation that the modern notion of number has been derived.
对于斯多葛派 ( Stoicorum Veterum Fragmenta [ SVF ], II, p. 328)，自然既是将宇宙的组成部分结合在一起的力量，也是地球生物的生成原理。该原则包含在精子的逻辑中，即一切事物所依赖的开创性逻辑，有生命的和无生命的，它的对应物是统一的，近似无理数的关系序列由此产生。12
For the Stoics (Stoicorum Veterum Fragmenta [SVF ], II, p. 328), nature was both the force that holds the constituent parts of the cosmos together and the generating principle of earthly beings. The principle was contained within the spermatikòs lógos, the seminal lógos on which everything depended, animate and inanimate, and which had its counterpart in the unity from which the sequences of relations that approximate irrational numbers emanate.12
对于新柏拉图主义者来说，根据柏拉图关于自然如何符合数字级数的观察（Epinomis，990 e），将数字与phýsis严格相关是司空见惯的。physikòs arithmós的概念见于 Plotinus ( Enneads , VI, 6, 16, 45-6) 和 Gerasa 的 Nicomachus (公元1-2 世纪), 13对他们来说，自然phýsis是由数字和根据精确的算术定律组织的生成流所涉及的关系。在 11 世纪，拜占庭僧侣 Michael Psellus 提出了物理数的概念（ physikòs lógos ) 作为数学数字的补充概念。前者更接近于活体、植物和动物，因为它们中的每一个都在确定的时间周期中出生、生长和死亡。14
For the Neo-Platonists as well it was commonplace to think of numbers in strict correlation to phýsis, in accordance with Platonic observations on how nature conformed to numerical progression (Epinomis, 990 e). The concept of physikòs arithmós is found in Plotinus (Enneads, VI, 6, 16, 45–6) and in Nicomachus of Gerasa (first–second century AD),13 for whom nature, phýsis, was a fabric made up of numbers and relations involved in generative flows organized according to precise arithmetical laws. In the eleventh century the Byzantine monk Michael Psellus would come up with the idea of physical number ( physikòs lógos ) as a complementary concept to mathematical number. The former pertains more closely to living bodies, plants and animals because each one of these is born, grows and dies in determined temporal cycles.14
所有这些宏大的学说复合体，系统地暗示数字和phýsis之间的密切关系，都能够建立在这样一个事实之上，即数字需要“根据晷针的性质 [ phýsin ]”来思考：a在当时，甚至在未来，这一因素将使一个抽象的数字模型和一个为数学描述自然而设计的算法系统变得可行。在几个世纪的过程中，“数字的真实性”不仅会回应noetós的哲学愿景宇宙，一个神圣的和可理解的宇宙，但也通过符号模型对自然进行真正的破译。在那之后的很长一段时间里，这个指针并没有继续揭示它的所有潜在功效。然而，在 Viète 和 Newton 之后，古希腊、印度和巴比伦算法所基于的相同范式用于定义更高级的计算科学。为了深入了解古代世界计算与知识之间的深层相互关系，仅靠哲学公式是不够的：我们还必须跟踪每一个计算细节，以捕捉其与这些公式的一致性和理论贡献。
All of this grandiose complex of doctrines, of systematic allusions to the affinity between numbers and phýsis, was capable of being founded, presumably, on the fact that numbers need to be thought ‘according to the nature [phýsin ] of the gnomon’: a factor that at the time, and even more so in the future, would render operative an abstract model of numbers and a system of algorithms devised for the mathematical description of nature. Over the course of centuries, the ‘reality of numbers’ would have answered not only to the philosophical vision of a noetós cosmos, of a divine and intelligible universe, but also to the real deciphering of nature through symbolic models. And for a long while after that the gnomon did not go on to reveal all of its potential efficacy. After Viète and Newton, however, the same paradigms on which the ancient Greek, Indian and Babylonian algorithms had been based served to define the structure of more advanced methods of the science of calculation. In order to penetrate into the deep interrelation between calculation and knowledge in the ancient world, philosophical formulas are not enough: we must also follow every computational detail to capture its conformity with and theoretical contribution to those very same formulas.
7. 中场休息：精神力学
7. Intermission: Spiritual Mechanics
两种关于灵魂本质的神秘而简洁的陈述从古代传给了我们。第一个归因于赫拉克利特：“灵魂属于自我增长的逻辑”（22 B 115 DK）；第二个是 Xenocrates，他在 Speusippus 之后被委托指导柏拉图学院：“灵魂是一个自动移动的数字”。亚里士多德 ( On the Soul , 408 b 34 ff.) 将 Xenocrates 的判断归入完全不合理的理论范畴，但他并不认为详细阐述他的反驳理由是多余的。除此之外，灵魂哲学与关于宇宙本质的综合理论之间的密切联系，在苏格拉底提出的一个问题中（斐德鲁斯）, 270 c)：“但是，如果我们将自然 [ phýsin ] 作为一个整体排除在外，你认为有可能充分了解灵魂的本性 [ phýsin ] 吗？” 答案显然是否定的，它间接关联了数字和灵魂，因为柏拉图习惯于肯定，在数字级数中，人们可以阅读phýsis，而灵魂的“本性”正是它的phýsis—— 换句话说，要辨别它在lógos的增长。
Two enigmatic and laconic statements on the nature of the soul have come down to us from antiquity. The first is attributed to Heraclitus: ‘to the soul belongs a lógos that increases itself’ (22 B 115 DK); the second to Xenocrates, who was entrusted with the direction of the Platonic Academy after Speusippus: ‘the soul is a self-moving number’. Aristotle (On the Soul, 408 b 34 ff.) consigned Xenocrates’ judgement to the category of theories that were completely irrational, but he did not consider it superfluous to excogitate at length on his reasons for refuting it. To this may be added the intimate connection between a philosophy of the soul and a comprehensive theory of the nature of the universe that is prefigured in a question raised by Socrates (Phaedrus, 270 c): ‘But do you think it possible to know the nature [ phýsin ] of the soul to any sufficient degree if we dismiss nature [ phýseos ] as a whole?’ The answer, self-evidently negative, indirectly correlates numbers and the soul, because Plato was wont to affirm that in numerical progressions one could read the phýsis, and the ‘nature’ of the soul was precisely its phýsis – in other words, to discern its growth in the lógos.
赫拉克利特的话是不可翻译的，并且可以进行各种解释，这在很大程度上取决于赋予单词lógos的众多含义中的哪种含义。但在色诺克拉底写的明显荒谬的句子中，对于赫拉克利特来说，包含的lógos可以用更严格的术语来说是数字。在这两种情况下，无论是赫拉克利特的逻辑或Xenocrates的数字，都不是静止不动的：第一个增加；第二个动作。现在，kínesis（运动）对于亚里士多德来说是一个词，指的是任何一种过程，在此过程中，事物可能会改变方面、性质或位置。争辩说，运动也可能指代数，指有节制的运动，指以数字记录的时间流逝，这并非不恰当。我们知道，在lógos的最早含义中，有一个是指通过枚举进行选择和调查。
Heraclitus’ words are untranslatable and lend themselves to various interpretations, depending largely on which meaning, from among its many meanings, is given to the word lógos. But in the apparently absurd sentence penned by Xenocrates, what for Heraclitus comprised the lógos can be, in more circumscribed terms, number. In both cases, be it the lógos of Heraclitus or Xenocrates’ number, neither stands still: the first augments; the second moves. Now kínesis (movement) is a word that for Aristotle refers to any kind of process during the course of which a thing may change aspect, nature or position. It is not inappropriate to contend that kínesis might also refer to numeration, to a measured movement, to a passing of time registered by numbers. And we know that among the earliest meanings of lógos there is the one that refers to selecting and surveying by way of enumeration.
对于斯多葛派来说，灵魂的激情经常与自然相称（ phýsis ）。他们被认为遵循或更频繁地侵犯了lógos，也就是说，数字度量。对于 Chrysippus 来说，“反对自然”意味着“反对根据自然的直接逻辑[ katà phýsin lógon ]”（SVF，III，第 94 页）。对于 Citium 的芝诺 ( SVF , I, p. 51) 激情是无法衡量的灵魂扩张或收缩 ( álogos )，好像自然增长所依赖的数学定律，正是柏拉图看到的有效地印在数字级数中的自然增长，被违反了。为了把握数字、灵魂和phýsis的相互关联，如果我们撇开自然一词在几个世纪，特别是在 17 世纪所获得的含义的变化，我们可以求助于斯宾诺莎的论点，即“自然总是一样的，它的美德和行动的力量在任何地方都是一样的”（伦理学，III，序言）。
For the Stoics the passions of the soul were regularly commensurate with nature ( phýsis ). They were thought to follow or, more frequently, to infringe on the lógos, that is to say, numerical measure. ‘Against nature’ meant, for Chrysippus, ‘against the straight lógos according to nature [katà phýsin lógon ]’ (SVF, III, p. 94). For Zeno of Citium (SVF, I, p. 51) passion was an expanding or contracting of the soul beyond measure (álogos ), as if the mathematical laws on which natural growth depended, precisely the natural growth that Plato saw effectively imprinted in numerical progressions, had been violated. To grasp the reciprocal affinity of number, soul and phýsis, if we leave aside the change in meaning that the term nature acquired during the course of centuries, and especially in the seventeenth century, we could resort to Spinoza’s thesis that ‘nature is always the same, and its virtue and power to act is everywhere one and the same’ (Ethics, III, Preface).
另一个特点将赫拉克利特和希诺克拉底的句子结合在一起：数字自己移动；lógos自我增强。运动和成长的发生是由于灵魂的内在倾向，即通过连续的扩展来独立地成长和改变自己的状态，就像在种子的发育中一样。布鲁诺·斯内尔1指出这种精神自主发展的可能性对于荷马来说仍然很陌生，荷马总是将身体和精神力量的增长归因于神灵的干预。在荷马眼中同样陌生的是质子 kinoûn的概念，这是起源于灵魂本身的主要原因，正如亚里士多德所设想的那样。
Another peculiarity brings together the sentences of Heraclitus and Xenocrates: number moves by itself; the lógos augments itself. Movement and growth occur due to the intrinsic predisposition of the soul to grow and to change its own state independently, by means of successive expansions, just as in the development of a seed. Bruno Snell1 noted that this possibility of autonomous development of the spirit was something still quite alien to Homer, who always attributes growth in physical and spiritual force to the intervention of a deity. Equally alien in Homer’s eyes would be the notion of prôton kinoûn, the primary cause originating in the soul itself, as Aristotle would conceive of it.
但是，这个自我移动的灵魂的“来自自身”是从什么产生的呢？种子在开始时是不可见的，几乎是机械的，依靠一种虚拟的力量，而这种力量的起源尚未向我们透露。Simone Weil 谈到了“精神力学，其中的定律虽然不同，但与力学本身的定律一样严格”。2这里提到马可 (4, 26-32)，将我们不知道的自动成长的王国 ( mekýnetai , automáte ) 比作在地球上发芽的种子。但正如柏拉图指出的那样，灵魂也与身体联系在一起（Phaedon, 83 d-e)，每一种快乐和痛苦都将灵魂固定并几乎钉在肉体上，使其相信一切都必须是真实的，身体声称是真实的。如果顺着这种倾向，它就不能纯粹地到达阴间，而是会落入另一个身体并根据其本性生长：“就像种子一样，它会在那里发芽[ hósper speiroméne emphýesthai ]，因为这永远不会参与到神圣、纯洁、统一的事物中”。
But from what is this ‘from itself’ of the self-moving soul generated? The seed grows invisibly at the start, almost mechanically, by dint of a virtual force the origin of which has not been disclosed to us. Simone Weil speaks of a ‘spiritual mechanics, the laws of which, though different, are as rigorous as those pertaining to mechanics itself’.2 There is a reference here to Mark (4, 26–32), where the Kingdom growing automatically (mekýnetai, automáte ) and unknown to us is compared to a seed germinating in the earth. But the soul is also tied to the body, as Plato pointed out (Phaedon, 83 d–e), and every pleasure and pain fixes and almost nails down the soul to a corporeal form, inducing it to believe that everything must be true that the body claims is true. If it were to follow this inclination, it could not then arrive pure in Hades but would fall into another body and grow according to its nature: ‘as if it were a seed, it will germinate there [hósper speiroméne emphýesthai ], and because of this will never participate in that which is divine, pure, uniform’.
这让我想起了 Epicharmus 的话，关于成长是如何受制于规律的——这些话又与歌德的歌词有显着的相似性：
This puts me in mind of the words of Epicharmus on how growth occurs subject to a law – words which in turn have a remarkable affinity with one of Goethe’s lyrics:
就像它带你来到这个世界的那一天
Just as on the day it brought you into the world
太阳升起，向行星致意，
the sun was up above greeting the planets,
很快你就长大了
soon you grew bigger
根据规范你开始的法律。
according to the law that regulated your beginning.
所以你一定是，你无法逃避自己——
And so you must be, you cannot escape yourself –
古代的先知和女巫如是说；
so said the Prophets and Sibyls of old;
时间和力量都无法打破
neither time nor strength can break
贯穿一生的印记形式。3
the imprinted form that develops throughout life.3
当它们由迭代公式生成时——从最简单的（包括每次将一个单位添加到一个自然整数以找到下一个整数）到最复杂的递归程序——数字，就像灵魂的情况一样，都是受到自动运动的影响。戴德金德和图灵都掌握了算术过程的自动本质，第一个是通过递归机制，第二个是对机械的更具启发性和明确的概念。尽管如此，如果不是戴德金德，那么至少图灵意识到了数字无法控制增长的潜在可能性。实际上，他理解如果计算的数字变得非常大，机器的算术误差会如何增长，并且他量化了误差的增长，在线性方程组的情况下，某些矩阵的大小。
When they are generated by an iterative formula – from the simplest, which consists of adding a unit to a natural integer every time to find the next one, to the most complex recursive procedures – numbers, just as in the case of the soul, are subject to an automatic movement. Dedekind and Turing would both grasp the automatic nature of arithmetical processes, the first through the mechanism of recursion, the second with a more suggestive and explicit notion of the mechanical. Nevertheless, if not Dedekind then at least Turing was aware of the insidious possibility of an uncontrollable growth of numbers. He understood, in effect, how an error in the arithmetic of the machine can grow if the numbers being calculated become very big, and he quantified the growth in error, in the case of a system of linear equations, by mean of functions capable of measuring the magnitude of certain matrices.
如果自然，正如尼采所指出的那样，真的是过度的，“浪费到无法衡量的程度，冷漠到无法衡量的程度，没有目的和顾忌”，4如果事实上，这种过度的性格属于phýsis—— 海德格尔也将其归结为一种压倒性的、过度涌入的存在对它来说——那么可以想象，在其他方面类似于phýsis的数字也可能超出所有控制。随着数字的增长，不仅无限即将出现：通过大规模自动微积分，我们的目光必须学会在它之前停止，在有限的坚定且仍然鲜为人知的基础上，无论是非常大还是非常小的。
If nature, as Nietzsche noted, is really excessive, ‘wasteful beyond measure, indifferent beyond measure, without purpose and scruple’,4 if, in truth, this excessive character belongs to phýsis – an overwhelming, excessive influx of presence that Heidegger also attributed to it – then it is conceivable that numbers, which resemble phýsis in other respects, may also grow beyond all control. There was not just infinity on the horizon with the growth of numbers: with large-scale automatic calculus our gaze had to learn to stop well before it, on the firm and still little-known ground of the finite, be it very big or extremely small.
8.芝诺悖论：运动的解释
8. Zeno’s Paradoxes: The Explanation of Movement
关于无限的不可知性的最矛盾的启示起源于埃利亚的芝诺，以辩证段落的形式出现，几个世纪以来从未真正被驳倒过。芝诺悖论的历史是由相互矛盾的评论、批评和反驳组成的；重新审视和令人惊讶的辩护最终将其意义颠倒过来：芝诺是对的，世界需要根据他的论点重新思考，如果我们阐述一个连贯且适当的连续统数学理论，悖论就会显现出来成为重新诠释现实世界的理想工具。它们不是无意义的推测，它们是现实，它们展示了数学可以表示它的方式。这至少，
The most paradoxical revelation of the unknowable nature of the infinite originates with Zeno of Elea, in the form of dialectical passages that throughout the centuries have never really been refuted. The history of Zeno’s paradoxes is made up of contradictory commentaries, critiques and refutations; of revisitations and surprising vindications that have ended up turning their meaning upside down: Zeno was right, the world needs to be rethought according to his arguments, and if we elaborate a mathematical theory of the continuum which is coherent and appropriate, the paradoxes reveal themselves to be ideal instruments for reinterpreting the real world. They are not idle speculation, they are reality, and they show the way in which mathematics can represent it. This, at least, is the explanation that seemed to prevail in the early twentieth century, with the development of a mathematical theory of the continuum, the origins of which date back to the second half of the nineteenth century.
关于运动主题的悖论是基于所覆盖距离的逐渐减少，以及观察到可以划分连续线的部分的无限减少导致世界在我们看来是不真实的东西. 似乎无限（ápeiron ）早在公元前 6 世纪和 5 世纪就作为一个困境进入了 Eleatic 思想，因为它以双重版本表达：一个是肯定的，尤其是在萨摩斯岛的梅利苏斯；另一个负面的，显然与第一个相反，就像在芝诺的悖论中一样。梅利苏斯的表达方式与巴门尼德不同。对后者而言，无限不是存在的属性，因为在他看来，它被一个非存在所否定，即希腊语中隐含的缺席（stéresis ）。虽然对于巴门尼德来说，必然性的力量似乎将存在限制在限制的束缚中（28 B 8, 26-33 DK），但梅利苏斯坚定地断言，一直存在并将永远存在的存在，没有开始或结束，是无限的——并且没有什么有开始和结束的可以称为无限的（30 B 1-4 DK）。无限，梅利苏斯经常断言，也是一（30 B 5 DK），因为如果它是二它会在另一件事上有限制（ pròsállo ）。
The paradoxes on the subject of movement are based on the progressive diminution of the amounts of distance covered, and on the observation that the decrease ad infinitum of the segments in which a continuous line can be divided causes the world to appear to us as something unreal. It seems that the infinite (ápeiron ) entered into Eleatic thought as far back as the sixth and fifth centuries BC as a dilemma because it is expressed in a double version: one affirmative, above all in Melissus of Samos; the other negative and apparently in opposition to the first, as in Zeno’s paradoxes. Melissus expresses himself differently from Parmenides. For the latter the infinite is not an attribute of being, because in his view it is negatively marked by a non-being, the absence (stéresis ) implicit in the Greek ápeiron. While for Parmenides the force of necessity seems to constrain being within the fetters of limitation (28 B 8, 26–33 DK), Melissus resolutely affirms that the being that has always been and will always be, without beginning or end, is infinite – and that nothing that has a beginning and an end can be called infinite (30 B 1–4 DK). The infinite, Melissus was wont to assert, is also one (30 B 5 DK), because if it were two it would have a limit in another thing ( pròs állo ).
从芝诺的角度来看，无限是产生悖论的引擎，但结果却截然不同：否认运动的可能性，或者表明我们对感官感知的理性解释不足。
The infinite, from Zeno’s point of view, is an engine for generating paradoxes, with very different results: denying the possibility of movement, or showing the insufficiency of our rational explanations of what we perceive with our senses.
我们在阿喀琉斯的追逐中发现的二分法没有尽头。它是一种潜在无限的完美例子。如果阿喀琉斯必须穿过一米长的空间，他必须穿过长度递减的区间 ½, ¼, ⅛ ... ½ n，其中n假定为无限值。如果乌龟以最小的优势出发，在阿喀琉斯覆盖地面的时间里，乌龟会向前迈出一小步；如果按照同样的逻辑继续追击，那么阿喀琉斯将永远追不上乌龟。而且，从逻辑上讲，阿喀琉斯甚至连一米都无法覆盖，因为他必须先覆盖一半，然后再覆盖剩下的一半，也就是四分之一，以此类推，无止境. 如果对轨迹中的每个点都考虑到在到达该点之前需要覆盖的无限区间，那么就不得不得出结论，阿喀琉斯实际上根本没有移动。
The dichotomous division we find in Achilles’ chase has no end; it is the perfect example of a kind of potential infinity. If Achilles has to cross a space one metre in length, he must pass through the intervals of decreasing length ½, ¼, ⅛ … ½n, where n assumes infinite values. If the tortoise sets off with even the most minimal advantage, in the time it will take Achilles to cover that ground the tortoise will make a small step forwards; and if the pursuit is continued according to the same logic, then Achilles will never be able to catch up with the tortoise. What’s more, Achilles is not logically able to cover even a metre, because he must first cover half of it, then half of what remains, that is to say a quarter of the whole – and so on, ad infinitum. If one considers for every point in the trajectory the infinity of intervals that need to be covered before reaching that point, one is obliged to conclude that Achilles, in fact, does not move at all.
如果看一下运动定律和将覆盖的空间s与所用时间t和速度v 联系起来的简单公式—— 也就是说，s = tv—— 很明显，如果阿基里斯以常数运行速度v ，那么即使乌龟具有初始优势s 0 ，他也会以小于v的恒定速度v ´超越正在移动的乌龟。实际上，存在一个t值，其中tv（跟腱所覆盖的距离）优于tv ´ + s 0，也就是乌龟走过的距离。
If one looks at the laws of motion and at the simple formula that links the space covered s to the time taken t and to the speed v – that is to say, s = tv – it becomes apparent that, if Achilles runs with a constant speed v, then he will overtake the tortoise that is moving with a constant speed v´ that is less than v, even if the tortoise has an initial advantage s0. In effect, there exists a value of t whereby tv, the distance covered by Achilles, is superior to tv´ + s0, that is to say, the distance covered by the tortoise.
阿喀琉斯比乌龟获得明显优势的科学依据还源于这样一个事实，即轨迹被划分成的无限区间的总和s是有限的，恰好等于 1。如果阿喀琉斯的速度高于某个有限值v ，他将在不超过s/v = 1/ v的时间内完成一米长的路线. 但是，如果他的速度逐渐减慢，并像划分空间的间隔长度一样趋于 0，他将无法完成它。但是悖论仍然存在：阿喀琉斯是如何设法覆盖在有限时间内将一米的轨迹“分段”成的无限间隔？或者，更一般地说，一个人如何在有限的时间内执行无限数量的操作？基于部分区间之和是有限这一事实的论点为阿喀琉斯何时会超越乌龟这一问题提供了答案。但仍然需要澄清的是，他将如何做到这一点。他将如何跨越无限连续的区间。1情况似乎仍然如此，而且同样难以理解，为了覆盖一条直线，阿喀琉斯必须触及它的所有点。
A scientific basis for the obvious advantage that Achilles will gain over the tortoise also stems from the fact that the sum s of the infinite intervals into which the trajectory is divided is finite, being precisely equal to 1. If Achilles maintains a speed superior to a certain finite value v, he will complete the metre-long course in a time not superior to s/v = 1/v. He would not be able to complete it, however, if his speed diminished progressively and tended towards 0 like the length of the intervals into which the space is divided. But the paradox remains: how does Achilles manage to cover the infinite intervals into which a trajectory of a metre is ‘segmented’ in a finite time? Or, more generally, how can one execute in a limited time an unlimited number of operations? The argument based on the fact that the sum of the partial intervals is finite provides an answer to the question when will Achilles overtake the tortoise? But what still needs to be clarified is how he will manage to do this. How will he cross an infinite succession of intervals.1 It still seems to be the case, and equally incomprehensible, that in order to cover a straight line Achilles must touch all of its points.
在提到悖论时，亚里士多德会反思穿越距离所花费的时间（物理学，233 a 21-31；239 b 5-240 a 18）。在没有极限理论的情况下关于数值级数，他认为，最重要的是，一条连续的线可以通过两种方式被认为是无限的：与分割相关或与末端相关。2只有在第二种情况下，所花费的时间是无限的，而在前一种情况下，如果速度保持在一个有限阈值之上，它实际上是有限的。他的论点是合理的，但仍然缺乏通过观察课程运行的无限间隔之和是有限的并且等于 1 得出的结论。
In referring to the paradox, Aristotle would reflect on the time it took to cover the distance (Physics, 233 a 21–31; 239 b 5–240 a 18). In the absence of a theory of limits pertaining to numerical series, he argued that, above all, a continuous line can be considered infinite in two ways: in relation to division or in relation to the extremities.2 Only in the second case is the time taken infinite, while in the former, if the speed is sustained above a finite threshold, it is actually limited. His argument is sound, but it still lacks the conclusion that follows from observing that the sum of the infinite intervals of the course run is finite, and equal to 1.
芝诺提出了他反对运动存在的第三个论点，它与箭的飞行有关，如下：如果我们接受每个事物在占据与其自身相等的空间时处于静止或运动状态，并且物体只在瞬间移动，然后箭头保持不动。3亚里士多德（物理学, 239 b 5-9) 总结了这个论点，好像芝诺想到了一种原子结构的空间和时间，一个由无限不可分割的元素组成的连续统一体：“芝诺论证的谬误是显而易见的：正如他声称一个事物在它静止时是静止的。没有以任何方式从其体积所占据的空间中移动，并且在其整体运动的假定过程的每个固定瞬间，它都保持在它在那一瞬间占据的空间中，那么箭头在整个运动过程中的任何时刻都不会移动它的飞行过程。但这是一个错误的结论，因为时间不是由实际的瞬间组成的，就像其他任何数量都不是由原子元素组成的一样。
Zeno develops his third argument against the existence of motion, relating to the flight of an arrow, in the following terms: if we accept that each thing is in a state of stillness or movement when it occupies a space equal to itself, and that the object only moves in the instant, then the arrow remains motionless.3 Aristotle (Physics, 239 b 5–9) summarized the argument as if Zeno thought of an atomistic fabric of space and time, a continuum composed of infinite indivisible elements: ‘The fallacy of Zeno’s argumentation is obvious: as he alleges that a thing is still when it has not moved in any way from the space occupied by its volume, and in every fixed instant of the supposed course of its overall movement it remains in the space that it occupies during that instant, then the arrow does not move at any moment during the course of its flight. But this is a false conclusion, because time is not composed of actual instants, any more than every other quantity is composed of atomic elements.’
因此亚里士多德坚持认为，只有当我们假设时间由瞬间组成时，飞行中的箭头才会保持静止。箭在不可分的瞬间实际上是不可能移动的，因为在那一瞬间，通过移动，它会改变位置，这意味着这一瞬间是真正可分的。4但尚不清楚芝诺是否假定原子论假设是必要的给出了他悖论的逻辑。即使假设空间和时间不是分别由不可分割的点和瞬间组成的，并且假设划分的无限是完全假设的，运动仍然会构成一种不可理解的现象。箭头移动一秒钟的事实假定相同的箭头在前半秒飞行，在这半秒的前半秒，以此类推，无限期地。A. N. Whitehead 评论说，芝诺一定模糊地意识到，如果我们从整体上考虑这个过程并问自己是什么发生了变化，就不可能给出确切的答案。任何可能已经移动的东西都以在前面的时间间隔内移动的东西为前提，并且由于无法完成课程，我们被迫得出结论，没有任何东西实际上已经搬家了。5
Hence Aristotle maintains that the arrow in flight remains stationary only if we suppose that time is made up of instants. It is in fact impossible for the arrow to move in the indivisible instant, because in that instant, by moving, it would change position, which would mean that the instant is really divisible.4 But it remains unclear as to whether Zeno had assumed the atomistic hypothesis as a necessary given for the logic of his paradox. Even if one assumes that space and time are not composed, respectively, of indivisible points and instants, and that the infinite by division was entirely hypothetical, movement would still constitute an incomprehensible phenomenon. The fact that the arrow moves for a second presupposes that the same arrow flies for the first half-second, for the first half of this half-second, and so on, indefinitely. A. N. Whitehead remarked that Zeno must have been vaguely aware that if we consider the process in its entirety and ask ourselves what it is that has moved, it is impossible to give a certain answer. Whatever may have moved presupposes something that has moved in the preceding interval and, unable to complete the course, we are forced to conclude that nothing has actually moved.5
阿喀琉斯和乌龟悖论中隐含的一个问题，以及莱布尼茨和牛顿对无穷小分析的最早调查中争论的一个问题，涉及变量是否可以达到自己的极限。牛顿在《原理》（第一卷，第一节）中为不可分方法的有效性辩护，构想了“新生”和“消逝”的量，并认为在有限时间内收敛的量和量之间的关系以连续方式对等价关系比任何固定距离更接近彼此，最终变得平等。伯克利主教（Bishop Berkeley）在他著名的对数学使用无穷小的论战控诉中断言，数学科学最终未能提供许多人期望最终解释宗教奥秘的清晰而独特的想法。6
An issue implicit in the paradox of Achilles and the tortoise, and one debated from the earliest inquiries into analysis of the infinitesimal by Leibniz and Newton, relates to whether variables can reach their own limits. Newton in the Principia (Book I, Section I) defended the efficacy of the method of indivisibles, conceiving of quantities that were ‘nascent’ and ‘evanescent’, and argued that quantities and the relations between quantities that converge in a finite time in a continuous manner towards a relation of parity get closer to each other than any fixed distance, in the end becoming equal. Bishop Berkeley, in his celebrated polemical indictment of the mathematical use of the infinitesimal, asserted that mathematical science fails in the end to provide the clear and distinct ideas that many expect to eventually explain the mysteries of religion.6
罗素随后提供了一个决定性的贡献，表明芝诺关于运动的悖论是远非诡辩。1903 年，在他的《数学原理》（第 327 段）中，他写道：
Russell subsequently supplied a decisive contribution which indicated that Zeno’s paradoxes regarding motion were far from mere sophisms. In 1903, in his Principles of Mathematics (par. 327) he would write:
在这个任性的世界里，没有什么比死后的名声更任性的了。后人缺乏判断力的最著名受害者之一是 Eleatic Zeno。发明了四个论证，所有论证都极其微妙和深刻，后来哲学家的粗俗宣称他只是一个巧妙的杂耍者，他的论证是一个和所有的诡辩。经过两千多年的不断反驳，这些诡辩被恢复了，并为一位德国教授奠定了数学复兴的基础，他可能从未梦想过自己与芝诺之间有任何联系。Weierstrass 通过严格地排除所有无穷小，最终表明我们生活在一个不变的世界中，箭在飞行的每一刻都是真正的静止。7
In this capricious world, nothing is more capricious than posthumous fame. One of the most notable victims of posterity’s lack of judgement is the Eleatic Zeno. Having invented four arguments, all immeasurably subtle and profound, the grossness of subsequent philosophers pronounced him to be a mere ingenious juggler, and his arguments to be one and all sophisms. After two thousand years of continual refutation, these sophisms were reinstated, and made the foundation of a mathematical renaissance, by a German professor, who probably never dreamed of any connection between himself and Zeno. Weierstrass, by strictly banishing all infinitesimals, has at last shown that we live in an unchanging world, and that the arrow, at every moment of its flight, is truly at rest.7
罗素（《原则》，第 332 段）认为，关于箭头的论点阐明了一个简单的事实，而忽视这一事实已导致运动哲学陷入了数百年的泥潭。他对 Karl Weierstrass 的重新审视可以用以下方式解释：与 Augustin-Louis Cauchy 一起，Weierstrass 是第一个明确建立无无穷小分析的数学家，他断言函数f ( x ) 趋向于极限L，因为x趋于对l，如果对应于给定的正但仍然很小的值ε，则有可能得到一个正数δ（取决于ε），因此当x与l的距离小于 δ 时，f ( x ) 与L的距离小于ε。如果L = 0，当x趋于l时，函数f接近 0 ，但在定义中我们故意避免说f ( x ) 的值变得无穷小。因此，流动的概念，变量接近其极限的动态张力消失了，这仅仅是因为变量在ε和δ指定的范围内根本不移动，只假设它们适合它们的值。静止胜过运动。
Russell (Principles, par. 332) thought that the argument concerning the arrow articulated a fact that was simply elementary, and that the neglect of this fact had caused the philosophy of movement to be bogged down for centuries. His revisiting of Karl Weierstrass may be explained in the following way: along with Augustin-Louis Cauchy, Weierstrass was the first mathematician to clearly establish analysis without infinitesimals, asserting that a function f (x ) tends towards a limit L, for x that tends towards l, if, in correspondence with a given positive but nevertheless small value ε, it is possible to arrive at a positive number δ (dependent on ε) so that the distance of f (x ) from L is less than ε when the distance of x from l is less than δ. If L = 0, the function f approaches 0 as x tends towards l, but in the definition we deliberately avoid saying that the value of f (x ) becomes infinitesimal. Hence the idea of flow, of the dynamic tension of the variable towards its limit, vanishes, simply because the variables, within the confines designated by ε and by δ, do not move at all, assuming only the values that are proper to them. Immobility prevails over movement.
因此，我们可以将物体在瞬间t的速度定义为所覆盖的距离与覆盖它所花费的时间之间的关系的极限，该关系趋向于在瞬间t的时间变量。这个极限，一个简单的数字，是空间在t时刻作为旅行时间函数的导数。通过这种方式，可以避免在微积分的最初发展中设想的“渐逝量”。因此，罗素在《原则》 （第 447 段）中指出：
We can therefore define the speed of a body in an instant t only as the limit of the relationship between the distance covered and the time taken to cover it tending to the time variable at the instant t. This limit, a simple number, is the derivative of space as a function of travel time at the instant t. In this way, it was possible to avoid ‘evanescent quantities’ as conceived in the first developments of infinitesimal calculus. Thus Russell states in the Principles (par. 447):
可以看出，由于对无穷小的否定，以及对函数导数的相关纯技术观点，我们必须完全拒绝运动状态的概念。运动仅仅在于在不同的时间占据不同的地方，受制于连续性。. . 没有从一个地方到另一个地方的过渡，没有连续的时刻或连续的位置，没有速度之类的东西，除非是在一个实数的意义上，它是一组商的极限。拒绝将速度和加速度作为物理事实（即作为属于每个瞬间的属性）到一个移动点，而不仅仅是表示某些比率极限的实数），正如我们将看到的，在运动定律的陈述中存在一些困难；但是 Weierstrass 在微积分中引入的改革使得这种拒绝势在必行。
It is to be observed that, in consequence of the denial of the infinitesimal, and in consequence of the allied purely technical view of the derivative of a function, we must entirely reject the notion of a state of motion. Motion consists merely in the occupation of different places at different times, subject to continuity . . . There is no transition from place to place, no consecutive moment or consecutive position, no such thing as velocity except in the sense of a real number which is the limit of a certain set of quotients. The rejection of velocity and acceleration as physical facts (i.e. as properties belonging at each instant to a moving point, and not merely real numbers expressing limits of certain ratios) involves, as we shall see, some difficulties in the statement of the laws of motion; but the reform introduced by Weierstrass in the infinitesimal calculus has rendered this rejection imperative.
因此，有理数和无理数，被视为变量的界限，继承了实际和真实的性质速度和加速度等物理概念。在这些相同的数字中，可以识别直线上的点的原子实体。因此，运动只能通过时空坐标来解释，因此只能通过连续的固定和精确位置来解释。“力学只能通过静止来解释运动。” 8
Thus rational and irrational numbers, conceived as limits of variables, inherited the actual and real nature of physical concepts such as velocity and acceleration. In those same numbers it is possible to recognize atomic entities that are points on a straight line. Movement could be interpreted, then, only through the coordinates of space-time, and thus by way of successive fixed and precise positions. ‘Mechanics can only explain movement through immobility.’8
只有通过数字——这是重要的结论——才能发现时空连续体的真实性。可以执行此任务的数字可能是理性的，也可能是非理性的。此外，在魏尔斯特拉斯之后，实数（有理数和无理数）的存在似乎是数学家自由创造的结果，尽管是由数值语料库的客观属性引起的。在思想与自然、自由与现实之间，还有什么更好的契合呢？
It was only with numbers – and this was the important conclusion – that the reality of the spatio-temporal continuum could be found. And the numbers that could perform this task could be either rational or irrational. In addition, the existence of real numbers (both rational and irrational) would appear, after Weierstrass, to be the effect of the free creation of a mathematician, albeit one induced by the objective properties of the numerical corpus. What better accord could there be between thought and nature, between freedom and the actual?
怀特海解释说，客观世界总是在一方面是潜在的可分性，另一方面是相互关系和层次的双重方面表现出来，分割的过程在每一刻都表现为现实的现实。被感知的世界总是以其潜在的无限可分性出现，而定义现实客观的真实的原子实体存在于数学关系系统中。作为他观察的支持证据，怀特黑德重申了威廉詹姆斯的论点，他在思考芝诺时区分了立即或实际感知的世界的性质与我们的理性想象的无限可分性：
Whitehead explained that the objective world always expresses itself under the dual aspect of potential divisibility on the one hand and, on the other, of mutual relations and gradation that the process of division manifests, in every instant, as actual realities. The perceived world always appears in its potentially indefinite divisibility, while real, atomistic entities that define the realitas objectiva dwell in a system of mathematical relationships. As supporting evidence of his observations, Whitehead reiterated the thesis of William James, who when thinking about Zeno distinguished between the nature of a world immediately or actually perceived and the indefinite divisibility imagined by our reason:
要么你的体验没有内容，没有变化，要么是可感知的内容或变化。您的从字面上看，对现实的了解是通过感知的萌芽或滴滴而增长的。在理智上和反思后，您可以将它们分成组件，但正如立即给出的那样，它们完全或根本不存在。9
Either your experience is of no content, of no change, or it is of a perceptible amount of content or change. Your acquaintance with reality grows literally by buds or drops of perception. Intellectually and on reflection you can divide these into components, but as immediately given, they come totally or not at all.9
Whitehead 评论说，“连续性涉及潜在的东西，而实际是无法治愈的原子论”，10但由于康托尔和戴德金的理论，几何连续性已经被构想为实际数字的域。分析的算术设计已经原子化了连续扩展。最后，在数值连续统的理论中，真实性取决于定义的数值实体、真实划分系统的组成部分、瞬时事件与位于连续统上某个点的其他事件的关系。在数字和点之间，公理式地建立了双单义的对应关系，并且通过数字，空间中的点和时间上的瞬间获得了一种新的现实。正如罗素所写（原则，同级。326):
Whitehead remarked that ‘continuity concerns that which is potential, whilst the actual is incurably atomistic’,10 but geometric continuity had already been conceived of, thanks to the theories of Cantor and Dedekind, as a domain of actual numbers. The arithmetical design of analysis had already atomized continuous extension. Actuality depends, in the end, in the theory of the numerical continuum, on defined numerical entities, constituent parts of a system of real divisions, of instantaneous events in relation to other events situated at some point on the continuum. Between numbers and points a correspondence is axiomatically established that is bi-univocal, and by way of numbers the points in space and instants in time acquire a new species of reality. As Russell writes (Principles, par. 326):
在将自己限制在算术连续统一体中时，我们以另一种方式与常见的先入之见发生冲突。关于算术连续统一体，庞加莱先生公正地评论道：“如此构想的连续统一体只不过是按一定顺序排列的个体的集合，数量无限，它是真实的，但在彼此之外。这不是通常的概念，在这个概念中，连续体的元素之间应该有一种紧密的联系，使它们成为一个整体，其中点不在线之前，而是在线之前。观点。在著名的公式中，连续体是多样性中的统一，只有多样性存在，统一消失了。11
In confining ourselves to the arithmetical continuum, we conflict in another way with common preconceptions. Of the arithmetical continuum, M. Poincaré justly remarks: ‘The continuum thus conceived is nothing but a collection of individuals arranged in a certain order, infinite in number, it is true, but external to each other. This is not the ordinary conception, in which there is supposed to be, between the elements of the continuum, a sort of intimate bond which makes a whole of them, in which the point is not prior to the line, but the line to the point. Of the famous formula, the continuum is unity in multiplicity, the multiplicity alone subsists, the unity has disappeared.’11
对于庞加莱来说，连续统的数学概念是人类思维的产物，但它也是一种物理实验，有利于并使其几乎成为必然。连续统的密度是不同测量值之间简单比较的结果——例如，三个权重A、B和C的情况。可能发生A = B和B = C，因为实验上不可能区分A与B和B与C，但我们也有，实验上，A < C. 然后我们会倾向于相信，实际上，与实验本身的指示相反，A与B不同，B与C不同，并且其他权重将自己插入A和B之间以及B和C之间。这是基于非矛盾原则得出的结论，不依赖于用于测量重量的仪器的精度。
For Poincaré the mathematical notion of the continuum is the product of the human mind, but it is also a kind of physical experiment that favours and renders its creation almost a necessity. The continuum’s density appears as the consequence of a simple comparison between different measurements – as in the case, for instance, of the three weights A, B and C. It can happen that A = B and B = C, due to the experimental impossibility of distinguishing A from B and B from C, but that we also have, experimentally, A < C. We would then be inclined to believe that in reality, contrary to the indications of the experiment itself, A is different from B, and B from C, and that other weights insert themselves between A and B as well as between B and C. This is a conclusion based on a principle of non-contradiction that does not depend on the precision of the instruments used to measure the weights.
以类似的方式，似乎有必要强加所谓的连续统的完整性，也就是说，将所有坐标为有理数的点插入其框架中。事实上，有些曲线相交的点坐标不是有理数。一个例子是正方形的对角线和内切圆之间的交点。假设只存在有理坐标的点，我们将无法证明这个交点在现实中存在。如果我们在笛卡尔平面上画一个圆，以圆心为原点，半径等于边长为 1 的正方形的对角线，它与横坐标轴相交的点是距离原点的距离。由无理数
In an analogous way, it seems necessary to impose the so-called completeness of the continuum, that is to say, the insertion into its framework of all the points which have co-ordinates that are rational numbers. There are in fact curves that intersect at points with coordinates that are not rational numbers. An example would be the point of intersection between the diagonal of a square and the inscribed circle. Assuming that only the points with rational co-ordinates exist, we will not be able to argue that this point of intersection exists in reality. If we draw a circle on the Cartesian plane, with its centre as the origin and its radius equal to the diagonal of a square of side 1, this intersects with the axis of the abscissa at a point the distance of which from the origin is measured by the irrational number 2.12 Now it becomes impossible to suppress the tendency of our minds to consider the same point as equivalent to a real entity. Our mind cannot abide gaps in the continuum, and the most efficient way of avoiding them is to think of the coordinate points of intersection between lines as numbers that have the same reality as whole numbers and fractions. The number endows the point with reality, on the understanding that it is itself considered to be a real entity. As Russell explained, ‘infinity and continuity appear together in pure arithmetic’ (Principles, par. 435). It was this intellectual achievement that presented itself as a solution to the difficulty that Zeno raised regarding movement and the nature of the continuum. The modern solution of the Achilles paradox was based on assuming the reality or possibility of precisely that which Zeno considered paradoxical, that is to say, according to Russell’s observation, the absence of a state of motion: a sacrifice that salvaged an indispensable fact: the actual existence of things. An actual entity, Whitehead observed, does not move: it is where it is, and it is that which it is.13
罗素认为运动状态的想法是不合理的，因为运动是由在确定的瞬间占据的原子位置组成的，这两者都可以通过实数同等地访问，对应于直线上的点。亚里士多德已经证明（物理学，234 a 24 ff.）没有任何东西可以在瞬间移动（nŷn)，并且因为这个时间不是由瞬间组成的。罗素回答说，事实上，这是真的，没有任何东西在瞬间移动——这与由魏尔斯特拉斯、戴德金和康托尔详细阐述的欧几里得度量提供的算术连续统的连贯理论兼容。只有这样，才能保证变化和运动的事物的真实性。矛盾的变成了真实的。
Russell had argued that the idea of a state of motion is not sound, because movement is made up of atomic positions occupied in determinate instants, both of these being equally accessible through real numbers, corresponding to points on a straight line. Aristotle had demonstrated (Physics, 234 a 24 ff.) that nothing can move in the instant (nŷn), and that because of this time is not made up of instants. Russell replied that, in effect, this is true, that nothing moves in the instant – and that this is compatible with a coherent theory of the arithmetical continuum supplied by the Euclidean metric, as had been elaborated by Weierstrass, by Dedekind and by Cantor. Only in this way could one guarantee the reality of that which changes and moves. The paradoxical becomes the real.
尽管如此，在罗素的评论中，还是有可能发现一种强迫，一种解决某种冲突的努力在数学严谨性和常识之间。但是常识，正如其他数学理论的情况一样，不得不让位于基于公式证据的愿景。数学一直是一门悖论的艺术，它的公式经常引起负责发现或设计它们的科学家的怀疑。但数学也是一门艺术，它通过能够使我们认识到我们所期望的定义和理论，尽可能地构建我们共同概念的模拟和忠实模型。我们可以在罗素的评论中看到这种几乎难以察觉的紧张，随后是希尔伯特和伯内斯以及随后斯蒂芬克莱恩对芝诺关于动议的第一个悖论的评论中明确的、毫不含糊的尴尬：
In Russell’s commentary it is nevertheless possible to detect a kind of forcing, an effort to resolve a type of conflict between mathematical rigour and common sense. But the common sense, as also happened in the case of other mathematical theories, had to give way to a vision based on the evidence of formulas. Mathematics has always been an art of paradox, and its formulas have often provoked a reaction of incredulity in the very scientist responsible for having discovered or devised them. But mathematics is also an art of constructing, as far as possible, simulations and faithful models of our common conceptions, by means of definitions and theories capable of making us recognize what we expect. That quasi-imperceptible straining that we may discern in Russell’s comments is followed by the explicit, unambiguous embarrassment of the commentary relating to Zeno’s first paradox concerning motion by Hilbert and Bernays, and subsequently by Stephen Kleene:
这个悖论有一个更激进的解决方案。这包括认识到，我们绝不必须相信运动在空间和时间方面的数学表示对于任意小的空间和时间间隔在物理上是有意义的；相反，我们有充分的理由认为，这个数学模型是从某个经验领域推断事实，也就是说，在迄今为止我们感知的数量级范围内的运动，被理解为一个简单的概念构造，类似于连续体的力学产生了一种推断，在这种推断中，人们假设空间以连续的方式被填充，不是以任何方式给出的，而是通过智力程序进行内插或外推的。14
There is a much more radical solution to the paradox. This consists of recognizing that we are in no way obliged to believe that the mathematical representation of movement in terms of space and time is physically meaningful for intervals of space and time that are arbitrarily small; rather we have every reason to suppose that this mathematical model extrapolates facts from a certain domain of experience, that is to say movements within the orders of magnitude accessible until now to our perception, understood as a simple conceptual construction, analogous to the way in which the mechanics of the continuum effects an extrapolation in which one assumes that space is filled, in a continuous fashion, by matter … The situation is similar in all cases in which we believe that it’s possible to exhibit an [actual] infinity directly as a fact of experience or of perception … A more careful examination shows then how an infinity is not in any way given, but is interpolated or extrapolated by means of an intellectual procedure.14
但是，除了推断之外，别无他法，用连续统的数学模型完成从经验中得出的事实，正如赫尔曼·外尔所指出的那样，这反过来又可以简化为纯粹的象征性构造。亚里士多德（物理学, 263 a 25–30) 曾指出，如果连续体被反复分成两半，这不会导致关于线或运动的连续性。他强调，运动最恰当地与连续性有关，不可否认，其中有无限数量的一半——但只是潜在的，而不是在实践中。简而言之，我们可以用以下方式总结它：认为移动是通过计数来实现的，这是荒谬的。但后来很明显，直线的运动和连续性不能仅通过自然数找到解释，在此基础上，事物将被逐一列举。需要一种新的、更普遍的数论，以及扩展现实性或 entelechy 的概念以涵盖 19 世纪末将被称为的东西，实数。
But there was precisely no other path than that of extrapolating, of completing the facts derived from experience with a mathematical model of the continuum, which in turn could be reducible, as Hermann Weyl noted, to a mere symbolic construction. Aristotle (Physics, 263 a 25–30) had remarked that if the continuum is repeatedly divided into two halves, this cannot result in continuity with respect to line or movement. Movement, he emphasized, pertains most properly to the continuous, and in this there is undeniably an infinite number of halves – but only potentially, not in practice. Put simply, we could summarize it in the following manner: it is absurd to think that what moves does so by counting. But then it became clear that movement and the continuity of the straight line could not find an explanation through only the natural numbers, on the basis of which things would be enumerated one at a time. A new, more general theory of numbers would be necessary, as well as an extension of the idea of actuality or entelechy to encompass what at the end of the nineteenth century would come to be called, and not for nothing, the real numbers.
9. 多元化的悖论
9. The Paradoxes of Plurality
数学连续统的现代理论也试图解开和澄清芝诺关于多元概念的悖论。根据资料来源，多元悖论作为假设证明了没有大小的存在是不存在的，并且在每一个被赋予大小和密度的事物中，它的组成部分在某种程度上彼此远离（29 B 1 丹麦克）。现在，组成复数的部分，由无数个元素组成，可能有一个正的大小或一个为零的大小。在第一种情况下，如果各部分具有相等的大小，无论多么小，我们无论如何都会获得无限大的大小。在第二种情况下，我们的量级为零，因为无论是有限总数的一部分还是无限总数的一部分，单个零量级的总和总是零. 没有扩展的点的无限总和必须产生 0 的组合扩展。
Modern theories of the mathematical continuum have also sought to disentangle and clarify Zeno’s paradoxes on the concept of plurality. According to the sources, the paradox on plurality had as a supposition the demonstration that a being without magnitude does not exist, and that in each thing that is endowed with magnitude and density its component parts are in some way distant from one another (29 B 1 DK). Now the parts that make up a plurality, consisting of an infinite number of elements, may have a positive magnitude or one that is null. In the first case, if the parts have an equal magnitude, however small, we obtain in any event an infinite magnitude. In the second case we have zero magnitude, because whether part of a finite or an infinite total, the sum of individually null magnitudes is always null. An infinite sum of points without extension must produce a combined extension of 0.
如果通过将一条连续的线划分为无穷大，我们得到了不可分割的元素，那么我们将因此遇到一个悖论：如果不可分割的元素的长度相等且大于 0，则线的长度是无限的；如果这些元素的长度为 0，那么它们的集合的总长度也将为 0。因此，如果存在许多生物，则它们必须同时又大又小：大到足以具有无限的量级，并且小到完全没有量级”（29 B 1 DK）。
If by dividing a continuous line to infinity we reached elements that were indivisible, we would thereby encounter a paradox: if the indivisible elements have an equal length that is greater than 0, then the length of the line is infinite; if these elements have 0 length, their aggregate will also have a total length of 0. ‘Hence, if there are many beings, it is necessary that they should be at the same time both large and small: large enough to have an infinite magnitude, and small enough to lack magnitude altogether’ (29 B 1 DK).
在他反对巴门尼德学说的论战中，亚里士多德诉诸芝诺的论点，根据该论点或一个不可分割且没有大小的单位，无论加减都不会产生任何影响，因此构成了一个非实体，一个纯粹的虚无：“如果某物本身是不可分割的，那么根据芝诺的概念，它就是虚无。” 实际上，芝诺否认了任何不会通过增加或减少事物而使事物变大或变小的事物的存在——假设不言而喻的事实是真实的事物必须具有大小”（形而上学，1001 b 7；29 21 DK）。
In his polemic against the doctrine of Parmenides, Aristotle resorts to the thesis of Zeno according to which a point or a unit which is indivisible and without magnitude does not produce any effects whether it is added or subtracted, and consequently constitutes a non-entity, a pure nothing: ‘If something is in itself indivisible, then according to Zeno’s conception it is nothing. In effect, Zeno denies any existence to that which does not make a thing bigger or smaller by being added to or subtracted from it – assuming as self-evident the fact that what is real must have magnitude’ (Metaphysics, 1001 b 7; 29 A 21 DK).
因此，对于亚里士多德来说，芝诺认为，如果添加的东西不会产生增加，那就什么都不是，同样的道理也适用于通过减去不会导致减少的东西。这是一个重要的观察，可能与柏拉图关于现实本质的观念有关。对于柏拉图来说，现实是建立在dýnamis的基础上的，即产生任何类型效果的能力，因此特别是增长或减少。这一概念的核心是数字级数的发展与phýsis 固有的增长引擎之间的类比。同样，在芝诺的推理中，事物的真实性似乎取决于产生增长或缩小的能力。
Hence, for Aristotle, Zeno argued that if something added does not produce increase it is nothing, and the same goes for something that by being subtracted does not result in a decrease. This is an important observation that may be connected to the Platonic idea of the nature of reality. For Plato, reality was founded on the dýnamis, the ability to produce an effect of any sort and, consequently, in particular, a growth or a diminution. At the heart of this conception was the analogy between the development of numerical progressions and the engine of growth inherent in phýsis. Similarly, in Zeno’s reasoning, the reality of a thing seems to depend on the capacity for producing growth or diminution.
亚里士多德似乎接受了反驳巴门尼德的推理脉络，但宣称自己反对 0 度长度必然是不真实的论点：“不可分割的东西当然可以存在”（形而上学，1001 b 14）。事实上，亚里士多德承认几何点的存在，他通常称之为stigmé，这个术语暗指一种刺穿，因此暗指在空间中的实际插入。欧几里得、阿基米德和后来的作者倾向于使用术语semeîon。1 Proclus 将继续在这些点中找到“可理解的物质性”的内涵，并解释说该点“因为它出现在想象的深处”而变得物质化（欧几里得评论'元素'，96）。Dianoetic 智能为抽象单元分配空间中的位置，将其转换为一个点，在想象（ 幻想 ）中找到一个适当的形象，即使仅限于物质和虚假的存在。奥古斯丁也将继续在独白中记录真实形象（智力的对象）和由想象创造的虚构之间的区别，希腊人称之为幻想或幻想。
Aristotle seems to accept the vein of reasoning that refutes Parmenides, but declares himself in opposition to the thesis that a 0 degree of length is necessarily unreal: ‘Something that is indivisible can certainly exist’ (Metaphysics, 1001 b 14). In fact, Aristotle admits the existence of the geometric point, which he generally calls stigmé, a term that alludes to a kind of piercing, and hence to an actual insertion in space. Euclid, Archimedes and later authors tend instead to use the term semeîon.1 Proclus would go on to find in these points a connotation of ‘intelligible materiality’, explaining that the point becomes materialized ‘inasmuch as it appears in the recesses of the imagination’ (Commentary on Euclid’s ‘Elements’, 96). Dianoetic intelligence assigns to the abstract unit a position in space, commutes it to a point, finding in the imagination ( phantasía ) an adequate figure, even if limited to a material and spurious existence. Augustine, too, would go on to register, in the Soliloquies, the difference between a real figure (an object of the intellect) and a figment created by the imagination, which the Greeks called phantasía or phántasma.
如果亚里士多德确实将现实归因于几何点，那么他仍然否认该点作为物质存在，而当他提到原子论时，通常是为了反驳它们。这个点可以被定义为原子的，因为它是不可分割的，但不是因为它可以与有形的原子相提并论，因为根据亚里士多德“没有物体是不可分割的点”（在天堂, 296 a 17–18)。然而，这并不意味着对他来说，点与其他所有数学实体一样，具有与可感知实体分开的现实：“如果几何学处理的实体碰巧是感觉事物，它不会将它们作为感觉对象来研究，而数学科学将不是理性的科学；此外，它们不会是与可以用感官感知的事物分离的其他对象的科学”（形而上学，1078 a 1-5）。
If Aristotle does attribute reality to a geometric point, he nevertheless denies that the point exists as something that is material, and when he refers to atomistic theories it is usually in order to refute them. The point may be defined as atomic because it is indivisible, but not because it is comparable to a corporeal atom, for according to Aristotle ‘no body is an indivisible point’ (On the Heavens, 296 a 17–18). Nevertheless, this does not imply for him that points, as for every other mathematical entity, have a separate reality from the perceptible one: ‘If it happens that the entities which geometry deals with are sensory things, it does not study them as sensory objects, and mathematical sciences will not be sciences of the sensible; furthermore they will not be sciences of other objects separated from what can be perceived with the senses’ (Metaphysics, 1078 a 1–5).
亚里士多德的一个点与一个整体（monás ）的区别仅在于它在空间中的位置。对于亚里士多德来说，点是一个有位置的统一体，但线不是由点组成的，两点之间没有连续性，点不是线的一部分（论生成和腐败，317 a 10；物理学，215 b 19-20），正如在他看来时间不是由瞬间组成的。相反，线是由点通过运动产生的（在灵魂上，409 a 4），因此点的现实再次存在于它通过一种扩展或增长产生量级的能力中.
A point for Aristotle is distinguished from a whole (monás ) only by its location in space. For Aristotle the point is a unity that has position, but the line is not made up of points, there is no continuity between two points and a point is not part of a line (On Generation and Corruption, 317 a 10; Physics, 215 b 19–20), just as in his view time is not made up of instants. The line, instead, is generated by the point by means of a movement (On the Soul, 409 a 4), and so the reality of the point resides once again in its capacity to produce magnitude by way of a kind of extension or growth.
说一条线由点组成或由它们组成是什么意思？Zeno 的论点是由一个关于组合的直观概念支持的，即元素的加法或减法，但如果我们考虑到最近的理论赋予诸如总和、维度、聚合和数量等术语的不同含义，这个论点就会变得更加复杂. 整体的大小是通过它的维度还是通过它的基数（或功率）来衡量的？维度和基数不相互依赖，因为可以设想具有 0 长度但具有连续体基数的点的线性组合。康托尔的三元set 恰好具有这些特征：它由 0 到 1 之间的所有数字组成，这些数字具有以 3 为底的表示形式，其中永远不会出现等于 1 的数字（也就是说，数字是 0 或 2）。在几何上，整体是通过考虑区间 [0, 1] 并重复切割第一个和最后一个三分之一来获得的。
What does it mean to say that a line is made of points, or composed of them? Zeno’s argument is sustained by one single intuitive idea about composition, that of the summation or subtraction of elements, but the argument becomes more complicated if we take into account the different meaning more recent theories assign to terms such as sum, dimension, aggregate and quantity. Is the magnitude of the whole measured by its dimensions or by its cardinality (or power)? Dimension and cardinality do not depend on each other, because it is possible to conceive of linear combinations of points with 0 length but with the cardinality of the continuum. Cantor’s ternary set has precisely these characteristics: it consists of all the numbers between 0 and 1 that have a representation in base 3 in which a number equal to 1 never appears (that is to say, the number is either 0 or 2). In geometrical terms, the whole is obtained by considering the interval [0, 1] and repeatedly cutting the first and the last third.
通过这种方式，我们获得了包含在 0 和 1 之间的数字子集的无限连续，每个子集都由不连接的闭区间之间的并集组成。康托尔三元集是所有这些整体的交集，它不能用数字来表示，因为它与大于 0 小于 1 的实数集是双义对应的，但它是不连续的和处处被间隙打断（间隙集无处不在密集，因为在其中两个之间，无论多么靠近，总是有另一个）并且它的长度为空。因此，我们从中获得区间 [0, 1] 中的点组合，该组合具有连续统的幂且长度等于 0。
In this way we obtain an unlimited succession of subsets of the numbers included between 0 and 1, each of which consists of the union between disconnected closed intervals. Cantor’s ternary set, which is the intersection of all of these wholes, cannot be rendered in numbers, because it is in bi-univocal correspondence with the set of real numbers which are greater than 0 and less than 1, and yet it is discontinuous and everywhere interrupted by gaps (the set of lacunae is everywhere dense, because between two of these, however close together, there is always another) and its length is null. Hence we obtain from this a combination of points in the interval [0, 1] that has the power of the continuum and a length equal to 0.
然而，没有维度的对象，例如一条线的点，有可能形成一个更高维度的整体（线） - 也就是说，维度为 1 - 并且基于欧几里得度量连贯地定义数字连续统一体可以为没有长度的单个点的聚合分配大于 0 的长度。这样，在极值A和B之间的直线上的间隔处，我们分配了一个精确等于b – a的长度，其中a和b是对应于点A和B的实数。如果a与b不同, 这个长度是一个非 0 的数字，即使A和B所描绘的直线段是由没有延伸的点组成的。数值连续统的度量理论，如果被认为是连贯的，那么它的后果之一就是对支持芝诺关于多元性问题的悖论的推理的反驳。
Nevertheless, it is possible that objects without dimension, such as the points of a line, can form a whole (the line) of superior dimension – that is to say, of dimension 1 – and the Euclidean metric defined coherently on the basis of the numerical continuum can assign a length greater than 0 to aggregates of individual points with no length. In this way, at an interval on the straight line between extremes A and B we assign a length that is precisely equal to b – a, where a and b are the real numbers corresponding to the points A and B. If a is different from b, this length is a number other than 0, even if the segment of the straight line traced by A and B is made up of points without extension. The metric theory of the numerical continuum, if taken to be coherent, would have among its consequences a confutation of the reasoning that underpins Zeno’s paradox on the question of plurality.
即使我们假设宇宙是由连续的物体组成的，亚里士多德指出（论生成和腐败，325 a），如果宇宙是完全可分的，那么就没有统一性也没有多重性，一切都是空的——而且，我们不知道如何争辩说存在不可分割的扩展片段；它们将是纯粹的虚构。面对虚构的空白或传统假设的不一致的这种风险也有助于解释古代数字的吸引力是真实的基础。自从古代原子论虽然没有被亚里士多德接受，芝诺悖论的逃生路线却集中在现实的一个可能元素上，它包括数字及其关系，以及它们的关系。测量几何形状、等价和相似关系的能力。算术和几何之间、数字和直线上的点之间存在问题的张力，揭示了这个程序固有的所有困难。寻找现实以反对回归或无限发展的不确定性，尤其是留基普斯和德谟克利特发展的原子论背后的动力。德谟克利特曾提出物体的品质是习俗和惯例的产物：“颜色是惯例，甜味是惯例，苦味是惯例——而唯一真实的东西[ etehêi ] 是原子和虚空' (68 B 125 DK)。尽管如此，德谟克利特还争辩说“我们对真实一无所知：实际上，真相要在深处被发现”（68 B 117 DK）。
Even if we hypothetically propose that the universe is made up of contiguous bodies, Aristotle noted (On Generation and Corruption, 325 a), if the universe is completely divisible, there can be neither unity nor multiplicity, everything is void – and, moreover, we cannot know how to contend that pieces of extension exist that are indivisible; they would be pure fictions. This risk of facing the fictive void or the inconsistency of conventional assumptions also helps to explain the attraction of numbers in antiquity as the foundation of the real. Since the ancient atomistic theories, albeit not accepted by Aristotle, the escape route from Zeno’s paradoxes has focused on a possible element of reality that consists in numbers and their relations, and in their capacity to measure geometric shapes, the relations of equivalence and similitude. The problematic tension between arithmetic and geometry, between numbers and points on a straight line, reveals all the difficulties that are inherent in this programme. The search for actualities to stand against the indeterminacy of a regression or a progression ad infinitum must have been, in particular, a motivating force behind the atomistic theories developed by Leucippus and Democritus. Democritus had propounded the qualities of objects as being the product of custom and convention: ‘Colour is according to convention, sweetness is down to convention, bitterness is according to convention – while the only real things [etehêi ] are atoms and the void’ (68 B 125 DK). Despite this, Democritus also contended that ‘we know nothing of the real: in effect, the truth is to be found in the depths’ (68 B 117 DK).
后来，卢克莱修使用与芝诺相同的二分法论证，以证明原子的存在是真实完全溶解的唯一替代方案（《宇宙的本质》，I，615-27）：
Later on, Lucretius used the same argument about dichotomy as Zeno in order to demonstrate that the existence of atoms is the only alternative to the complete dissolution of the real (On the Nature of the Universe, I, 615–27):
此外，除非有一些最小的东西，
Besides, unless there is some smallest thing,
最小的身体将由无限的部分组成，
The tiniest body will consist of infinite parts,
既然这些可以减半，它们的一半又减半，
Since these can be halved, and their halves halved again,
永远，没有尽头的分裂。
Forever, with no end to the division.
那么这两者之间会有什么区别
So then what difference will there be between
所有事物的总和和最小的事物？
The sum of all things and the least of things?
根本不会有。因为虽然事物的总和
There will be none at all. For though the sum of things
将是完全无限的，最小的物体
Will be completely infinite, the smallest bodies
将同样由无限部分组成。
Will equally consist of infinite parts.
但既然真正的推理反对这一点，
But since true reasoning protests against this,
并告诉我们头脑无法相信它，
And tells us that the mind cannot believe it,
你必须承认失败，并承认
You must admit defeat, and recognize
存在根本没有部分的事物，
That things exist which have no parts at all,
自己最小。既然这些存在
Themselves being smallest. And since these exist
你必须承认它们组成的原子
You must admit that the atoms they compose
本身也是坚固和永恒的。
Are themselves also solid and everlasting.
与一种仍然广泛流行的信念相反，数学并不是从纯粹的抽象中诞生的，而是作为物理现实的抽象模型。最初，实在原则必须委托给一种数字原子论，以反对由连续统的无限可分性引起的非现实性。数字、关系和算法起源于与ápeiron的对比，到无限大和无限小。这在毕达哥拉斯的情况下很明显：他们的数字被设想为在空间中分离的点的有序序列，其方式似乎是为了找到一个以实际存在的方式存在的现有实体的宇宙。有效的; 不是一开始可能出现的抽象，相反，它是对事物本质的有效实现。
Contrary to a still-widespread belief, mathematics is not born out of pure abstraction, as an abstract model of physical reality. The principle of reality had to be entrusted, initially, to a sort of numerical atomism, in opposition to the irreality evoked by the unlimited divisibility of the continuum. Numbers, relations and algorithms originate in contradistinction to the ápeiron, to the infinitely big and the infinitely small. This is evident in the case of the Pythagoreans: their numbers were conceived of as ordered sequences of points separated in space, in a way that seems to be responding precisely in order to found a universe of existing entities existing in a way that is actual and effective; not an abstraction, as at first it might appear, but on the contrary an effective realization of the substance of things.
根据亚里士多德（形而上学，1080 b 16 和 1036 b 8；58 B 9 和 58 B 25 DK），毕达哥拉斯学派坚持认为所有敏感物质都是由数字组成的，整个宇宙都是由数字组成的，并且几何形状比如圆形和三角形应该被视为人类的肉和骨头，以及雕像的青铜或石头。柏拉图本人，亚里士多德提醒我们（形而上学，987 b 22；58 B 13 DK），认为数字是事物实体的原因，实际上等同于事物本身。Leucippus 和 Democritus 会遵循这条推理路线，而且实际上，亚里士多德很有帮助地指出原子论者认为，就像毕达哥拉斯学派一样，“存在于宇宙是由数字组成的，或者是从数字发展而来的”（在天堂，303 a 8-9）。构成数字（ stoicheîon ）的内在要素是统一，正如点构成直线一样。一和重点是小、简单、不可分割、无处不在的，这种情况促使古代哲学家将它们视为原则（形而上学，1014 b）。
According to Aristotle (Metaphysics, 1080 b 16 and 1036 b 8; 58 B 9 and 58 B 25 DK), the Pythagoreans maintained that all sensitive substances were made of numbers, that the entire universe was made up of them, and that geometrical shapes such as the circle and the triangle should be regarded in the same way as the flesh and bones of humanity, and as the bronze or stone of a statue. Plato himself, Aristotle reminds us (Metaphysics, 987 b 22; 58 B 13 DK), thought that numbers were the cause of the substance of things, and in fact equivalent to the things themselves. Leucippus and Democritus would have followed this line of reasoning and, indeed, Aristotle helpfully points out that the atomists thought, just like the Pythagoreans, that ‘everything that exists in the universe is made up of numbers, or develops from numbers’ (On the Heavens, 303 a 8–9). The immanent element that was constitutive (stoicheîon ) of numbers was unity, just as the point was constitutive of the straight line. The one and the point are small, simple, indivisible, everywhere manifest, a circumstance that induced ancient philosophers to treat them as principles (Metaphysics, 1014 b).
毕达哥拉斯学派和原子论者的这个纲领在 19 世纪的数学中得到了完善，当时魏尔斯特拉斯、康托尔、戴德金德和其他人发展了一个关于数值连续统的连贯理论，该理论由包括有理数和无理数的领域定义。因此，我们可以理解确定无理数的存在以及它们与自然整数具有相同的本体论地位是多么重要。
This programme of the Pythagoreans and the atomists was perfected by the mathematics of the nineteenth century, when Weierstrass, Cantor, Dedekind and others developed a coherent theory of the numerical continuum, defined by the field that includes rational and irrational numbers. We can therefore understand just how important it was to establish that irrational numbers exist and that they have the same ontological status as natural whole numbers.
10. 有限与无限：不可通约性与算法
10. The Limited and the Limitless: Incommensurability and Algorithms
为什么整数和算法，不像无理数和连续统的概念，有自己无可置疑的现实？如果我们遵循柏拉图式的标准来定义什么是真实的，什么不是，我们将不得不重新审视Philebus（16 c）中断言它的段落，这要归功于从更接近神灵的古人那里继承的知识， “我们声称的现实总是存在的，因为它们是由一和多构成的，它们在自身内部将有限和无限联系在一起”。这提供了一个与巴门尼德（Parmenides）相距不远的愿景，根据普罗克鲁斯（Proclus）的说法，他断言“存在仅就概念 [ eîdos ] 而言是单一的，而就我们的感官经验的证据而言，存在是多重的 [enárgeia ]' (29 A 15 DK)。
Why do whole numbers and algorithms, unlike irrational numbers and the concept of the continuum, have their own unquestionable reality? If we follow the Platonic criterion for defining what is real and what is not, we would have to revisit the passage in Philebus (16 c) in which it is asserted, thanks to knowledge inherited from the ancients, who were nearer to the gods, that ‘the realities that we claim always exist, given that they are constituted by the one and the many, have interrelated within themselves the limited and the limitless’. This offered a vision not far removed from that of Parmenides, who asserted, according to Proclus, that ‘being is one only as far as the concept [eîdos ] is concerned, and multiple instead as far as the evidence of our sensory experience goes [enárgeia ]’ (29 A 15 DK).
对于柏拉图来说，自然整数arithmós正是介于 ( metaxý ) 无限和一 ( Philebus , 16 d–e) 之间、介于极限和无限之间、介于前者和后者之间的那个。有了算法，一个人能够计算，也就是说，产生数字和数字之间的关系，这种产生是在两个相反的两极之间实现的，这实际上是一个，被认为是开创性的lógos和无限的，数字和不同大小的几何形状的无限增长。号码和关系是由单元生成的，可以无限扩展的序列，通过连续的除法操作，1从而形成构成现实的密集中间纹理，或它的充分表示。为了近似我们现在称为无理数的实体，在古代已经存在康托尔将继续称为基本序列的东西，分数序列在实直线上的一点收敛。但是对于希腊人来说，现实并不包括戴德金德所说的理性语料库的“部分”或“切割”，或者我们用符号表示的无限十进制数字的数字
For Plato the natural whole number, the arithmós, is precisely that which comes between (metaxý ) the infinite and the one (Philebus, 16 d–e), between the limit and the limitless, that which mediates between the former and the latter. With algorithms, one was in a position to calculate, that is to say, to produce numbers and relations between numbers, and this production was realized between two opposite poles, which were in effect the one, conceived as the seminal lógos, and the limitless, the indefinite growth of numbers and of geometric shapes in different sizes. Number and relations were generated by the unit in sequences that could be extended infinitely, by way of successive operations of division,1 thereby forming a dense intermediate texture that constituted reality, or an adequate representation of it. To approximate the entities that we now call irrational numbers there already existed in antiquity what Cantor would go on to term fundamental sequences, sequences of fractions converging at a point on the real straight line. But for the Greeks reality did not consist of that which Dedekind would come to call a ‘section’ or ‘cut’ of the rational corpus, or in the number with infinite decimal digits that we denote, for example, with the symbol 2. That was an inexpressible lacuna, not an entity to which an actual physical existence could be attributed. Hence they lacked a precise idea of numerical approximation: there was no entity to approximate, and consequently it wasn’t even conceivable that there was a distance between the fraction and the limit of the sequence. The reality of the relations between geometrical magnitudes that were incommensurable, such as the diagonal and the side of a square, resided entirely in the relations between whole numbers, in the lógoi effectively calculated, and in the law that allowed them to be calculated. However, this circumstance in some cases did not impede the possibility of associating to a geometric line a whole number capable of measuring it approximately, even in the case when the line, for instance, could be the diagonal of a square with side 1. For Plato (Republic, 546 c) the rational diagonal of 5, that is to say 7 = 49, could be compared to 50, that is to say, the exact length of the diagonal of the square with a side 5. The number 7 = 49 is a whole number, while 50, which is an irrational number, constituted a lacuna for the Greeks, an entity that cannot be represented by means of integers.
柏拉图的论证是基于毕达哥拉斯定理应用于两边长度为 5 的直角三角形。因为这些边上的平方和等于斜边上的平方，从等价 5 2 + 5 2 = 50由此得出斜边的长度为
Plato’s argument is based on Pythagoras’ theorem applied to a right-angled triangle with both sides of length 5. Because the sum of the squares on these sides is equal to the square on the hypotenuse, from the equivalence 52 + 52 = 50 it follows that the length of the hypotenuse is 50. The square root of 49 is an approximation of it, which has the advantage of coinciding with a natural whole number, 7, instead of an irrational number such as 50. The constructions using ropes and wooden pegs on the Vedic fire altars as described in the Śulvasūtra make use of a similar approximation, that is to say: 52 + 52 ≈ 49.2 In any event, the measurement of the length of all bodies is never exact, and normally consists, as the text has it, of something only ‘more or less’ correct.3
近似的概念，在古代的数值程序中已经预示，最终将成为数学思想的基石之一。自远古时代以来（例如，在吠陀文本中清楚地暗示了这一事实），人们就知道量不能精确测量，而只能通过近似值、过量和缺陷（超过和低于标记）来测量。
The concept of approximation, already foreshadowed in the numerical procedures of antiquity, would eventually become one of the cornerstones of mathematical thought. Since time immemorial (with clear allusions to the fact, for example, in the Vedic texts) it was known that quantities cannot be measured exactly, but only through approximations, by excess and defect (exceeding and falling short of the mark).
如何近似无理数的问题不仅仅涉及计算的应用方面。测量近似度的技术将成为与数学思维完全相同的技术。事实上，欧几里得关系的定义要归功于先前通过过量和缺陷进行数值近似的技术。我们对 Weierstrass 的极限定义反过来又要归功于数值算法的存在（其中相同的符号ε和δ出现在以前表示近似误差）；用于逼近函数的解析公式用于定义一般概念，例如计算系统的稳定性；Joseph Liouville 在 1844 年证明的一个定理（见下文第 158 页）确立了通过逼近代数方程的根可获得的逼近程度，甚至允许量化自动计算中的关键现象，也就是说，计算的数字的增长速度。
The question of how to approximate irrational numbers did not merely relate to applied aspects of calculation. The techniques for measuring degrees of approximation would become the very same techniques of mathematical thought. In fact, the Euclidean definition of relation is indebted to prior techniques of numerical approximation by excess and defect; the definition of limit we owe to Weierstrass is in turn indebted for its logic to the existence of numerical algorithms (in which the same symbols ε and δ appear that in previous times had denoted the error of approximation); the analytical formulas used for approximating a function serve to define general concepts, such as the stability of a system of calculation; a theorem demonstrated in 1844 by Joseph Liouville (see p. 158 below) established the degree of approximation that was attainable by approximating the roots of an algebraic equation, and even allows the quantification of a crucial phenomenon in automatic calculation, that is to say, the speed of growth of the numbers calculated.
关于在欧几里得之前有可能确定不可通约几何量级存在的工具，我们可以冒险提出一些猜想。我们应该立即指出，对角线和正方形边的不可通约性的不同证明是可能的，并且计算有理近似的各种程序是可能的。
We can venture a few conjectures regarding the tools that made it possible, before Euclid, to determine the existence of incommensurable geometric magnitudes. We should point out immediately that different demonstrations of the incommensurability of the diagonal and the side of a square were possible, and various procedures for calculating rational approximations of 2. But we invariably find in these the theme of the growth and decrease of quantities. In one of the possible demonstrations the same dichotomous criterion used by Zeno appears, even if Aristotle seems to cast doubt on the possibility of such an alignment between incommensurability and the generative techniques behind the paradoxes (Prior Analytics, 65 b 16–21). The dichotomous method in any case leads to the construction of a limitless succession of squares, and in this way the age-old paradigm was repeated: building from an initial nucleus, formed by a geometric figure, to an infinite number of shapes similar to the first. Let’s assume, by reasoning ad absurdum, that the diagonal and side are commensurable, so that we may construct a sequence of squares of decreasing scale, by means of progressive halvings, associating each time with the diagonal and the side, as a consequence of the initial hypothesis of commensurability, a positive whole number. The linear segments become progressively smaller, as do the corresponding numbers. But the sequence of diagonals and sides is limitless, whereas the decreasing size of the numbers in the sequence cannot be: hence the absurdity.4
图 5
Figure 5
如果我们荒谬地假设（图 5）正方形 EFGH 的对角线 EG 与边 EF 可公度（在图中，E、F、G 和 H 是正方形 ABCD 边的中间点），它应该可以将关系 EG:EF 表示为两个整数d 和s (Euclid) 之间的关系。现在，正方形 ABCD 和正方形 EFGH 分别代表d 2和s 2 - 如图所示，其中一个是另一个的两倍。因此，d 2是偶数，并且由于奇数的平方是奇数，AB 也是偶数。这意味着它的一半 AF 代表一个整数。现在 EF 代表一个整数，因此 EF 和 AF 之间的关系是整数之间的关系。因此，我们构造了一个正方形 EKFA，它的边是正方形 ABCD 边的一半，所以它的直径和边代表整数。该构造可以无限地重复，生成渐进式的正方形更小，每次边都是前边的一半，但总是等于整数。数字变得越来越小，就像正方形一样，但是虽然包含连续大小的正方形可以无限减少，但整数不能小于 1。数字的无限减少与无限减半是不相容的行，因为该过程必须以统一的段或以奇数衡量的段结束。因此，二分法提供了一个很好的例子，说明了数字和几何图之间的那种不相容性，这种不相容性将继续导致发现不可通约的数量。
If we suppose, absurdly (Fig. 5), that the diagonal EG of the square EFGH is commensurable with the side EF (in the figure, E, F, G and H are intermediate points of the sides of the square ABCD), it should be possible to express the relation EG:EF as a relation between two whole numbers d and s (Euclid). Now, the square ABCD and the square EFGH represent, respectively, d 2 and s 2 – one of which, as shown in the diagram, is the double of the other. Therefore, d 2 is an even number and, given that the square of an odd number is odd, AB is also even. This implies that its half AF represents a whole number. Now EF represents a whole number, and because of this the relation between EF and AF is a relation between whole numbers. Consequently, we have constructed a square EKFA, the side of which is half of the side of the square ABCD, so that its diameter and its side represent whole numbers. The construction may be repeated indefinitely, generating squares that are progressively smaller, with the side each time being half the size of the preceding one but always equal to a whole number. The numbers become increasingly small, just as the squares do, but while the squares, which comprise continuous magnitudes, may be reduced indefinitely, the whole numbers cannot become inferior to 1. The indefinite decreasing of the numbers is incompatible with the indefinite halving of the lines, because the process should necessarily conclude in a unified segment or one measured by an odd number. The dichotomous method thus offers a perfect example of that kind of incompatibility between numbers and geometrical diagrams that would go on to lead to the discovery of incommensurable quantities.
二分法并不是研究不可通约性的唯一可能的权宜之计。另一个与前一个具有某种相似性的演示，甚至可以放在不可通约性研究的起源处，它与算法紧密相关，该算法可以在达到所需精度的情况下近似对角线和正方形的一侧。这两个过程，演示和算法，就像一枚硬币的两个面——每个都是另一个的反面——仍然可以用不定数量的正方形来说明。演示程序的基础是antanaíresis或anthyphaíresis，称为“欧几里得算法”的连续减法过程，该技术通过计算商和余数来找到两个量值之间的最大公分母。给定一个正方形 ABCD（图 6），在其对角线 BD = d上标记一个等于其边l的线段 BE 。此操作只不过是d除以l，其结果是 1，余数等于 ED。因此，将余数 ED 作为后续较小正方形的边l '，小于d的一半，然后从边l '和对角线的正方形开始构造d ′ = DF = l第三个正方形用同样的程序。该构造可以无限重复，并且在每个正方形 Q 中，对角线d等于边加上后续较小正方形 Q '的边的总和，即d = l + l '。5
The dichotomous method was not the only possible expedient device for studying incommensurability. Another demonstration with some affinity to the preceding one, which could even be placed at the origin of the study of incommensurability, was strictly tied to the algorithm that makes it possible to approximate, up to the required precision, the relation between the diagonal and the side of a square. The two procedures, demonstration and algorithm, are like two sides of the same coin – each is the reverse of the other – and can still be illustrated with the construction of an indefinite number of squares. The basis of the demonstration procedure is antanaíresis or anthyphaíresis, the process of successive subtractions known as the ‘Euclidean algorithm’, the technique for finding the largest common denominator between two magnitudes with the calculation of quotients and remainders. Given a square ABCD (Fig. 6), one marks on its diagonal BD = d a segment BE equal to its side l. This operation is nothing other than the division of d by l, the result of which is 1 with a remainder equal to ED. One takes therefore the remainder ED as the side l ′ of a subsequent smaller square, smaller than half of d, and proceeds to construct from the square of side l ′ and diagonal d ′ = DF = l a third square with the same procedure. The construction can be repeated indefinitely, and in every square Q the diagonal d is equal to the sum of the side plus the side of the subsequent smaller square Q′, that is d = l + l ′.5
图 6
Figure 6
推理是由荒谬发展而来的。如果大正方形的对角线 BD 和边 BC 是可公度的，则两者都是同一度量单位 U 的精确倍数。因此也可以得出差 BD – BC = EF 和差 DC – FC = DC – EF = DF 都是同一测量单位 U 的倍数（因为它们是 U 的倍数之间的差异）。也就是说，小正方形的边和对角线是 U 的可公度和精确倍数。构造越来越小的正方形，每条对角线和每条边都成为 U 的倍数。但这些倍数总是更小：如果 BC = 40U ，例如，则最大值为 EF = 39U。每边都等于n U，n总是较小的正整数。但是n不能小于 1，而平方可以无限减小。因此，结论是正方形的对角线和边不可公度（元素，X，2）。6
The reasoning develops by absurdity. If the diagonal BD and the side BC of the larger square are commensurable, then both are exact multiples of the same unit of measurement U. So it also follows that the difference BD – BC = EF and the difference DC – FC = DC – EF = DF are both multiples of the same unit of measurement U (because they are differences between multiples of U). That is to say that the side and diagonal of the smaller square are commensurable and exact multiples of U. Constructing ever smaller squares, every diagonal and every side becomes a multiple of U. But these multiples are always smaller: if it were BC = 40U, for example, then it would be a maximum of EF = 39U. Every side would be equal to nU, with n always being a smaller positive whole number. But n cannot become smaller than 1, whereas the squares can diminish indefinitely. Consequently, the conclusion is that the diagonal and side of a square are not commensurable (Elements, X, 2).6
现在必须注意的是，反向轨迹，从较小的正方形到逐渐变大的正方形，是通过与前一个完全相反的变换来实现的：两个变换由两个变换来识别两行两列的矩阵，其中一个是另一个的逆矩阵。7随着逐渐增加的边l和对角线d的连续性，我们构建了关系d : l近似
Now what one must pay attention to is that the inverse trajectory, in increasing scale from a smaller square to squares that are progressively larger, is realized with a transformation that is precisely the reverse of the preceding one: the two transformations are identified by the two matrices of two rows and two columns of which one is the inverse of the other.7 With the succession of progressively increasing sides l and diagonals d we construct the relations d :l that approximate 2. We thereby obtain, that is, an algorithm, a process of calculation of rational numbers (the sequence of fractions d/l ) that, as l increases, that is to say as 1/l decreases, gets progressively closer to 2 (without ever reaching it). It is the same algorithm of lateral and diagonal numbers that Theon of Smyrna (first to second century AD) refers to in his Expositio rerum mathematicarum ad legendum Platonem utilum. A clear demonstration of the co-presence of the ‘big’ and the ‘small’ as the Platonic designation of the ápeiron : the numbers d and l become increasingly bigger while the unit of measurement 1/l becomes increasingly smaller. Also, the distance of the relation d :l from 2 becomes increasingly smaller with the growth of l, but it can never be 0, and this could very well have represented a revealing argument for understanding that the diagonal and the side of a square are incommensurable.8
我们可以断言 2 正是关系d 2 : l 2连续的极限。事实上，我们可以这样计算l，使得 2 与关系d 2 : l 2之间的距离小于任意小的量ε，因为距离等于d 2 : l 2 – 2 = ± 1/ 2 . _ 这意味着关系的继承在接近 2 时变得越来越密集：当一个任意小的分数ε是固定的（例如ε = 1/ n ，其中n等于一个非常大的数）时，存在无限数量的关系d 2 : l 2与 2 的距离小于ε，而只有有限数量的关系与 2 的距离大于 ε。就分数而言，我们可以说分数序列d 2 / l 2 收敛于 2，或者说d /l 收敛于
We can assert that 2 is precisely the limit of the succession of relations d 2 :l 2. In fact, we can calculate l in such a way that the distance between 2 and the relation d 2 :l 2 is less than an arbitrarily small quantity ε, because the distance is equal to d 2 :l 2 – 2 = ± 1/l 2. This means that the succession of relations becomes increasingly dense in proximity to 2: when a fraction ε which is arbitrarily small is fixed (for example ε = 1/n , with n equal to a very big number), there exists an infinite number of relations d 2 :l 2 that are distant from 2 by less than ε, while only a finite number of relations are distant from 2 by more than ε. In terms of fractions, we would say that the sequence of fractions d 2 /l 2 converges at 2, or that d/l converges at 2.
这种构想极限的方式，连同柯西和魏尔斯特拉斯的现代表述，在某种意义上是违反直觉的，但它回答了一个更精确的数学标准。在连续关系或分数d / l的情况下 我们现在可以重复已经说明的关于函数极限的内容。直觉上，我们倾向于在动态意义上看待分数序列的展开，就好像它是一个趋向于接近目的地或目标点的分数运动的问题。但是，正如 17 世纪大多数人在分析无穷小时所设想的那样，动态展开在数学中没有找到明确的表述，并且在有效达到极限方面留下了模棱两可的空白。最严格的定义颠覆了这里的观点：我们首先选择一个任意小的区域，以限制L（半径为ε) 并在此基础上确定从哪一点起，继承的一般结束明确地在该区域内。根据 Cauchy 和 Weierstrass 的极限概念，由与动态张力相反的概念塑造，与数字的静态和原子概念一致，并与将无限理解为实际而非潜在的概念相关联。
This way of conceiving the limit, alongside the modern formulation of Cauchy and Weierstrass, is in a certain sense counter-intuitive, but it answers to a more precise mathematical criterion. In the case of the succession of relations or fractions d/l we could now repeat what has already been stated in relation to the limit of a function. Intuitively, we would be inclined to regard the unfolding of the sequence of fractions in a dynamic sense, as if it were a question of the movement of a fraction that tends to get closer to a point of destination or target. But the dynamic unfolding, as it was conceived by the majority in the analysis of the infinitesimal in the seventeenth century, does not find a clear formulation in mathematics and leaves equivocal gaps regarding the effective attainment of the limit. The most rigorous definition overturns the perspective here: we first select an area that is arbitrarily small for the limit L (of radius ε) and determine on the basis of this from which point onwards the generic end of the succession is definitively in that area. The same conception of the limit according to Cauchy and Weierstrass, shaped by an idea opposed to that of a dynamic tension, is aligned with a static and atomistic conception of numbers and correlates to an idea of the infinite understood as actual rather than potential.
或许值得指出的是，现代对数列极限的定义所提供的这种视角颠覆，与近似的逻辑密切相关。结石。在Cauchy 和 Weierstrass 之前的代数计算中，符号ε通常代表通过在某个端点（距解的距离ε处）停止的分数序列计算无理数时的残差。可以以固定确定端点的标准的方式预先分配误差ε 。类似地，在分数序列d 2 /l 2中，当残差 (1/ l 2 ) 小于预设分数ε时，我们可以决定停止该过程。9
It is perhaps worth pointing out that this overturning of perspective afforded by the modern definition of the limit of a sequence closely follows the logic of approximate calculus. In the algebraic computatio that preceded Cauchy and Weierstrass the symbol ε typically stood for the residual error in the calculation of an irrational number by means of a sequence of fractions stopped at a certain endpoint (at a distance ε from the solution). The error ε could be pre-assigned in such a way as to fix a criterion determining the endpoint. Similarly, in the sequence of fractions d 2/l 2 we can decide to arrest the procedure when the residual error (1/l 2) is less than a pre-set fraction ε.9
关系序列d 2 : l 2定义了一系列间隔I 1，I 2，...多余的（四舍五入）。间隔包含在另一个中，并在 2 左右逐渐变小，但它们的长度永远不会为空。对关系d : l也有类似的结论。但在这种情况下，区间包括一个不同于 2 的实体，不是一个整数，而是一个平方等于 2 的数字，用符号表示
The sequence of relations d 2 :l 2 defines a sequence of intervals I1, I2, … Ik, …, each one of which includes the number 2 and has as its extremities one approximation by defect (rounding down) and one by excess (rounding up). The intervals are included one within the other and become progressively smaller around 2, but their length never becomes null. A similar conclusion obtains for the relations d :l. But in this case the intervals include an entity of a kind different from 2, not a whole number but a number the square of which is equal to 2, designated by the symbol 2. Now it becomes impossible to avoid the perception of the actual presence of this entity, even though irrational, but one is obliged to consider as really existing, in opposition to the indefiniteness of the ápeiron, only the relations d :l. One may then define the number 2 as the same sequence of infinite intervals that include it, because the extremities of these intervals are the rational numbers that are actually capable of being calculated and that are precisely equal to the relations d :l. In such a case we may be inclined to substitute an individual entity for the graduality of approximation, a symbol that corresponds to a new number and denotes the sequence of intervals.
如果我们现在将任何无理数视为包含它的无限序列的区间，并且我们用符号表示它，我们被引导将符号数字的新域构想为无限的独立个体的多样性。这是一个决定性的步骤，可以在这些符号数中正式定义算术，类似于有理数。
If we now think of any irrational number as an indefinite sequence of intervals that includes it, and we denote it with a symbol, we are led to conceive of a new domain of symbol-numbers as a multiplicity of infinite separate individuals. This is a decisive step, facilitated by the possibility of formally defining an arithmetic, among these symbol-numbers, similar to that of rational numbers.
以下也是一个决定性因素：一个包含在另一个中的不定区间序列让位于将所有有理数的集合分成两类。第一类包含所有在区间I k左侧的有理数，对于足够大的k，也就是说，它小于所有区间I k的最左端，其中有限数量的例外。第二类包含优于所有区间I k的右手极值的有理数, 除了有限数量的这些。在这两个类中，我们添加了由所有区间中包含的有理数组成的第三类。现在第三个类，如果它不为空，则包含一个有理数r。如果第三类为空，即不包含任何有理数，则区间序列表示一个无理数。
The following is also a decisive factor: the indefinite sequence of intervals included one within the other gives way to a separation of the set of all the rational numbers into two classes. The first class contains all the rational numbers that are on the left-hand side of the intervals Ik, for a k that is sufficiently large, that is to say, which are less than the extreme left of all the intervals Ik, with the exception of a finite number of them. The second class contains rational numbers that are superior to the right-hand extremity of all the intervals Ik, with the exception of a finite number of these. To these two classes we add a third class formed by the rational numbers contained within all the intervals. Now the third class, if it is not empty, contains a single rational number r. If the third class is empty, that is to say, does not contain any rational number, the sequence of intervals represents an irrational number.
有理数集的这种划分正是戴德金德在 19 世纪末将其定义为有理语料库的一部分。实数，有理数和无理数，将被视为部分。但是截面的概念已经隐含在希腊数学中，确切地说是在antanaíresis的机制和比率d / l的构造中。推理类似于庞加莱在《科学与假设》（1902）中提出的推理：我们几乎觉得有必要设想一个数字，尽管它是非理性的，就像我们觉得有必要承认交点P是存在的和真实的一样在两条曲线之间笛卡尔平面，通过有理横坐标从右到左越来越接近P的横坐标。
This division of the set of rational numbers is precisely what Dedekind, at the end of the nineteenth century, would define as a section of the rational corpus. The real numbers, the rational and the irrational, would then be conceived as sections. But the concept of section is already implicit in Greek mathematics, precisely in the mechanism of antanaíresis and in the construction of the ratios d/l. The reasoning is similar to that proposed by Poincaré in Science and Hypothesis (1902): we almost feel obliged to conceive of a number, albeit irrational, exactly in the same way that we feel obliged to recognize as existent and real the point of intersection P between two curves on the Cartesian plane, getting closer from right to left through the rational abscissas increasingly close to the abscissa of P.
令人惊讶的是，在我们计算横向数和对角数的同一递归公式中，同时存在不存在的证明（一种常见的度量m不存在）以及有效构造在有理场中不存在的有理逼近。这是 Proclus 所区分的数学的两个方面：一个是理论类型，另一个是问题类型。第一个，试图建立一个理论结果，例如存在或不存在一个合理的解决方案；第二，如果不存在，则有效地寻求构建解决方案或近似它的分数。这是两种形式的互补推理，它们以完全相同的公式表现出来。
It may appear surprising to find, in the same recursive formula with which we calculate the lateral and diagonal numbers, the simultaneous presence of a demonstration of non-existence (a common measure m does not exist) and the effective construction of rational approximations of that which, in the rational field, cannot exist. These are two aspects of mathematics distinguished by Proclus: one theoretical in kind, the other of a problematical type. With the first, one attempts to establish a theoretical result, such as the existence or non-existence of a rational solution; with the second, one effectively seeks to construct the solution or a fraction that approximates it, if it does not exist. These are two forms of complementary reasoning that manifest themselves in the very same formulas.
越来越大或越来越小的正方形的构建是一个没有限制的过程。然而，即使没有最后一步，程序的形式和规律性也使我们能够就整个正方形序列和无限可能的选择得出结论性的论点。这其中隐含着逻辑上所谓的全称量词：任何线段，无论多么小，都不能是对角线和正方形边长的共同度量。我们有一个不存在的证明，但同时我们也通过几何证明接触到一个现实。现实与存在并不重合；它们是两个不同的概念，在不同的层面上运作。10现实是必然的东西，而不是基于任何约定俗成的东西。这种惯例会歪曲它的本质，削弱由公式和算法本身强加的必然性和禁令方面。我们不确定任何事情；是数字按照自己的规律排列，最终独立于我们。
The construction of increasingly large or increasingly small squares is a process without limit. Nevertheless, even in the absence of the last step, the form and regularity of the procedure allow us to reach a conclusive thesis regarding the entire succession of squares and an infinity of possible choices. Implicit within this there is what in logic is called the universal quantifier: any segment, however small, cannot be a common measure of the diagonal and the side of a square. We have a demonstration of non-existence, but at the same time we also touch a reality, by way of a geometrical proof. Reality and existence do not coincide; they are two distinct concepts that operate on different planes.10 Reality is something that is asserted by necessity, not on the basis of any kind of convention. Such conventionality would falsify its nature, weakening that aspect of inevitability and injunction that is imposed by the very formulas and the algorithms themselves. We do not determine anything; it is the numbers that arrange themselves according to their own law, which is ultimately independent of us.
边l和正方形对角线d的连续性，按升序计算，是无限的——但关系d : l的连续性有一个极限，用我们所说的“2 的平方根”表示，因为关系d 2 : l 2收敛于 2。11关系和极限的概念，与离极限的距离逐渐减小的近似计算相结合，使我们能够包含ápeiron的不确定性。即使它是开放的和有潜力的，ápeiron可以说，是由极限完善的，并且服从发展的规律，由此得出最终的结论。在关系的数学概念中，我们找到了ápeiron不确定性的解毒剂，并且在数字d和l的渐进生成中，可以“一下子抓住无限”，因为无限受到被抓住的关系的限制一次行动”。12这正是柏拉图在Philebus (16 c) 中所说的：可知且真实的实体是有限与无限的结合。
The succession of sides l and of the diagonals d of squares, taken in ascending scale, is infinite – but the succession of the relations d :l has a limit, represented by what we call ‘the square root of 2’, because the relation d 2:l 2 converges at 2.11 The concepts of relation and limit, joined to the calculation of approximations whose distance from the limit progressively decreases, allow us to encompass the inconclusiveness of the ápeiron. Even if it is open and potential, the ápeiron is, so to speak, perfected by the limit, and obeys a regularity of development from which final conclusions are reached. In the mathematical concept of relation we find the antidote to the indefiniteness of the ápeiron, and in the progressive generation of numbers d and l it is possible to ‘grasp the limitless at a stroke, because the limitless is limited by a relation that is grasped in a single action’.12 This is precisely what Plato says in Philebus (16 c): an entity that is knowable and real is a combination of the limited and the limitless.
在 Iamblichus、士麦那的席恩和 Proclus 中，也就是说，在最深入算法细节的原始资料中，没有明显的明确提到关系d 2 : l 2近似于 2 的事实。它有有人指出，柏拉图暗指“5 的有理对角线”（字面意思是可表达和可表达的对角线），即数字 7，我们知道它是边为 5 的正方形对角线的近似值，相反，这是不合理的，或者更好的是，无法形容和无法表达。但是，如果不是在理性和非理性的对比中，我们就不会遇到近似的概念。只有在政治家（287 c）的一段中，陌生人才从Elea 提到了diaíresis，指的是寻找连续分裂的策略，他说“一个人必须总是尽可能地按照最接近二的数字进行划分”——几乎就像解剖“牺牲的受害者”一样。这种接近（在希腊语中，engýtes ）可能暗示着距离的缩小，以及趋向于无理数的关系的密集增加。无理数如
In Iamblichus, in Theon of Smyrna and in Proclus, that is to say in the original sources that go deepest into the detail of algorithms, there is no discernible explicit reference to the fact that the relations d 2:l 2 approximate 2. It has been pointed out that Plato alludes to the ‘rational diagonal of 5’ (literally, the effable and expressible diagonal), that is to say, the number 7, which we know to be an approximation of the diagonal of the square with a side 5, which is on the contrary irrational or, better still, ineffable and inexpressible. But we do not encounter the concept of approximation if not in the contrast between rational and irrational. Only in a passage of the Statesman (287 c) does the Stranger from Elea refer to diaíresis, to the strategy of searching for successive divisions, saying that ‘one must always divide, as far as possible, according to the number closest to two’ – almost in the same way as a ‘sacrificial victim’ is dissected. This proximity (in Greek, engýtes ) may allude to the narrowing of distances, to the dense accretion of relations that tend towards irrational numbers. The irrational numbers such as 2 will be defined formally, in the nineteenth century, as sequences of rational numbers p/q (where p and q are whole numbers) that become denser the closer they get to their limit.
数字 7 被解释为“5 的有理对角线”，暗示了通过有理数来表达非理性的可能性。在数字（正整数）之间的关系中，我们发现了我们只能用负数来表达的东西的真实、具体和实现的表达，因为对于一个共同的测量单位来说，大小之间的关系是不存在的。士麦那席恩的评论强化了这种情况，即统一 ( monás )，它是边l和对角线d的初始值，在此基础上的关系d : l被构造，“作为万物的原则，必须具有边和对角的潜力[ dynámei ]”。13在dýnamis中，正如柏拉图所坚持的 ( Epinomis , 990 c)，我们发现了数字实体的现实，可以用它来表达无法表达的东西。
Interpreted as the ‘rational diagonal of 5’, the number 7 alludes to the possibility of expressing the irrational through the rational by means of numbers. In the relations between numbers (positive whole numbers) we find the real, concrete and actualized expression of that which we will only be able to express in negative terms, since the relation between magnitudes for a common unit of measurement does not exist. This is a circumstance that is reinforced with the comment by Theon of Smyrna whereby the unity (monás ), which is the initial value of both side l and diagonal d, on the basis of which the relations d :l are constructed, ‘as the principle of all things, must have the potentiality [dynámei ] of both the side and the diagonal’.13 In the dýnamis, as Plato insisted (Epinomis, 990 c), we find the reality of numerical entities with which it was possible to express the inexpressible.
在希腊数学和吠陀计算中已经很明显的一个突出事实是，等价关系以及几何图形的放大或缩小操作通常与数值大小上的类似递归操作密切相关。这些操作使定义分数p/q的整数p和q增长，并且分数近似于关系在无法衡量的量级之间；反过来，这种不可通约性在相似图形的增长或下降序列的极限行为中是可见的。
The salient fact, already evident in Greek mathematics and Vedic calculations, will be that relations of equivalence and the operation of enlargement or reduction of geometrical figures are generally in close relation to analogous recursive operations on numerical magnitudes. These operations make the whole numbers p and q that define the fractions p/q grow, and the fractions approximate the relations between incommensurable magnitudes; and the incommensurability is visible, in turn, in the behaviour at the limit of sequences of similar figures that grow or decrease.
现在，众所周知，近似值随着分数的分子p和分母q的增长而提高，这一特征已经可以在随递归定律增加的横向和对角数中追溯。但是计算器，无论是人类的还是数字的，总是在数字前加上一个数字t，并用超过t个数字对数字进行四舍五入，当接近极限时会造成无法弥补的信息损失。由于这个和其他原因，数字计算器更喜欢将分数p/q表示 为“浮点数”。14但这种权宜之计是不够的，因为正如我们将看到的那样，数量的增长会损害稳定性15程序，这是其有效性的基本要求。
Now, it is well known that the approximation improves with the growth of the numerator p and the denominator q of a fraction, a characteristic that is already traceable in the lateral and diagonal numbers that increase with a recursive law. But a calculator, whether human or digital, always operates with a number t prefixed to the numerals and rounds the numbers with more than t figures, causing irreparable losses of information as it nears the limit. For this and other reasons, the digital calculator prefers to represent the fractions p/q as ‘floating points’.14 But this expedient is not enough, because as we shall see the growth of numbers can compromise the stability15 of the procedures, which is a fundamental requirement for their effectiveness.
正如我们所见，不可通约性与连续统的结构有关。中心问题始终是连续体是由不可分割的部分组成，还是由又可以无限分割的部分组成。在论文De lineis insecabilibus（“论不可分割的线”）的一段中，包含在亚里士多德语料库（970 a 14 ff.）中，证明了不可分割线概念的不一致。一个人设想了一系列由antanaíresis调节的方格，欧几里得算法计算对角线和边的最大公分母。如果有人荒谬地假设有一条不可分割的线，这将是正方形的边，其对角线除以边，结果会得到一条长度小于边的线——否则，在对角线上构建的正方形将至少是前者的四倍而不是前者的两倍。然后会找到一条更小的线比边，假定是不可分割的，因此是荒谬的。这就是为什么亚里士多德可以断言原子论的理论不受数学家能够证明的任何事物的支持的原因之一，包括人们想象的对角线和正方形边的不可通约性。
Incommensurability, as we have seen, has to do with the structure of the continuum. The central question is always whether the continuum is made up of indivisible parts, or of parts that in turn may be divided indefinitely. In a passage of the treatise De lineis insecabilibus (‘On Indivisible Lines’), included in the Corpus Aristotelicum (970 a 14 ff.), the inconsistency of the concept of the indivisible line is demonstrated. One envisages a succession of squares regulated by antanaíresis, the Euclidean algorithm for the calculation of the highest common denominator of diagonal and side. If one were to suppose, absurdly, that there was an indivisible line, this would be the side of a square the diagonal of which, divided by the side, would as a result give a line with a length inferior to that of the side – otherwise, the square constructed on the diagonal would be at least four times rather than just twice as large as the former. One would then find a line smaller than the side, assumed to be indivisible, hence the absurdity. This is one reason why Aristotle could assert that atomistic theories are unsupported by anything mathematicians are able to demonstrate, including, one imagines, the incommensurability of the diagonal and the side of a square.
通过无限的相似图形序列来表示数字d和l的可能性在整个现存文献中都得到了体现，尽管具有不同的重点和几何含义。生成数字d和l的公式似乎与代数公式的变体有关，该变体对应于欧几里得 ( Elements , II, 10) 的定理，从中可以提取与平方序列有关的构造定律. 这表明，根据一个可疑且广泛讨论的论文，在《元素》中，我们不仅发现了一个几何代数，也就是说，用几何语言表达的代数，也是计算几何，一种数值演算，其结构与几何图形的属性相关。需要区分这两个方面，因为代数公式不是算法。例如，表示三个数的乘积的公式a × b × c对应于两个可能的过程，分别由两个表达式 ( a × b ) × c和a × ( b × c )，它可以给出两个完全不同的结果。16
The possibility of representing the numbers d and l by means of unlimited sequences of similar figures is signalled throughout the extant literature, albeit with differing emphases and geometrical implications. The formulas for generating the numbers d and l seem connected to a variant of the algebraic formula which corresponds to a theorem of Euclid’s (Elements, II, 10), from which it is possible to extract a law of construction relating to the succession of squares. This demonstrates that in the Elements we find not only, according to a dubious and widely discussed thesis, a geometrical algebra, that is to say, an algebra couched in geometrical language, but also a computatio geometrica, a numerical calculus the structure of which is correlated to the properties of geometrical figures. One needs to distinguish between these two aspects because the algebraic formulas are not algorithms. For example, the formula that denotes the product of three numbers, a × b × c, corresponds to two possible procedures, suggested by the two respective expressions (a × b) × c and a × (b × c ), which can give two completely different results.16
最后，有一个与吠陀几何有关的基本联系，即通过图 5 所示的构造实现的不可通约性的证明。在图 5 中存在四个相等的正方形，即 CHKG、BFKG、KFAE 和 HDEK。现在由这四个正方形组成的较大正方形与主要吠陀祭坛之一的主体重合，如图 1 所示。追踪祭坛中央主体每个正方形的对角线，并重复该操作以实现几何图形的渐进式嵌入或嵌套，最终证明了正方形的边和对角线的不可公度。同样的构造也导致了毕达哥拉斯定理应用于等边直角三角形的直观、直接的演示。据推测，关于建造火坛的吠陀论文的作者都知道这种演示。Albert Bürk 在他对 Āpastamba 的Shulba Sūtra ( Śulvasūtra ) 的评论中，没有不指出 Agni 的中心身体正是ātman，世界的无所不知的发生器，每个生物和普遍意识的核心，它在成长和形式的多样性之外仍然存在，总是与它自己相同和相同。17
There is in the end an essential nexus that links to Vedic geometry the demonstration of incommensurability realized by means of construction indicated by Figure 5. In Figure 5 there are four equal squares present, namely CHKG, BFKG, KFAE and HDEK. Now the bigger square composed of these four squares coincides with the body of one of the principal Vedic altars, as indicated in Figure 1. Tracing the diagonals in each one of the squares of the central body of the altar, and repeating the operation in order to realize a progressive emboîtement, or nesting, of geometrical figures, one ultimately demonstrates the incommensurability of the side and the diagonal of a square. The same construction also leads to the intuitive, immediate demonstration of Pythagoras’ theorem applied to equilateral, right-angled triangles. It is presumed that this demonstration was known to the authors of the Vedic treatises on the construction of fire altars. Albert Bürk, in his commentary on Āpastamba’s Shulba Sūtra (Śulvasūtra ), does not fail to point out that the central body of Agni is none other than ātman, the all-knowing generator of the world, the central nucleus of every living being and universal consciousness, which remains, beyond growth and the multiplicity of forms, always equal to and the same as itself.17
11. 数的实在：康托的基本数列
11. The Reality of Numbers: Cantor’s Fundamental Sequences
二十世纪末发展起来的算术连续统的数学理论似乎能够驳倒芝诺关于复数性质的悖论，该悖论依赖于一个假设，即非扩展量的有限或无限和，例如直线，总是一个零幅度。如果这个假设得到证实，那么现实就会消失在虚无中，消失在非存在中：这是一个不可接受的论点，在 Eleatic 逻辑中，它强加了多重性不存在而存在是一个的结论。. 巴门尼德关于存在是一体的、不动的、同质的、各向同性的和连续的，并且每一种多样性都是虚幻的，这一论点比任何多样性的存在所产生的后果更自相矛盾。
Mathematical theories of the arithmetical continuum developed at the end of the twentieth century seem to be capable of confuting Zeno’s paradox on the nature of plurality, which relies on the presupposition that a finite or infinite sum of non-extended magnitudes, such as points on a straight line, is always a null magnitude. If this presupposition were verified, then reality would vanish into the void and into non-being: an unacceptable thesis that, in Eleatic logic, imposed the conclusion that multiplicity does not exist and that being is one. Parmenides’ thesis that Being is one, unmoving, homogeneous, isotropic and continuous, and that every multiplicity is illusory, was still less paradoxical than the consequences that follow from the existence of any plurality.
数值连续统的理论首先基于直线上的点与实数（有理数，或分数，加上无理数）之间的双单义对应，这使我们能够将实直线视为几何和一个数字实体。这个实体可以用不同的、等效的方式定义。实数，也是一条直线上的一个点，是一个无限十进制数，既可以是周期性的，也可以是非周期性的。以等效的方式，实数由所谓的有理数（分数）的基本序列组成。他们的本体论地位在很大程度上用康托尔自己的话说：
The theory of the numerical continuum is based above all on the bi-univocal correspondence between points on a straight line and real numbers (rational numbers, or fractions, plus irrational numbers) that allows us to consider the real straight line as both a geometrical and a numerical entity. This entity is definable in different, equivalent ways. A real number, which is also a point on a straight line, is an infinite decimal number that is either periodic or aperiodic. In an equivalent way the real numbers consist of the so-called fundamental sequences of rational numbers (fractions). Their ontological status is in large part revealed in Cantor’s own words:
在根据定律获得无限系列有理数a 1 , a 2 , ..., a n , ... 的情况下，数值大小的概念有第一次推广，使得差a n+m -随着n的增加，无论正整数m的值如何， a n [绝对值] 变得无限小，或者换句话说，一旦分配了一个任意正有理数ε，一个整数n 1存在，对于该整数n+ m -一个n当n≥n 1时， [绝对值]小于ε，并且m是任意选择的正整数。我将这个系列的这个属性表达如下：'系列有一个确定的极限b '。因此，这些词仅用于阐明该系列的这一特性，而暂时没有提及另一个；并且当我们将系列与特定符号b联系起来时，我们必须以同样的方式将不同的符号b、b´、b"赋予同一物种的不同系列。1
There is a first generalization of the notion of numerical magnitude in the case where in virtue of a law an infinite series of rational numbers is obtained a1, a2, …, an, …, such that the difference an+m − an becomes infinitely small [in absolute value] as n increases, whatever the value of the positive integer m, or in other terms, such that, once assigned an arbitrary positive rational number ε, an integer n1 exists for which an+m − an [in absolute value] is less than ε when n ≥ n1, and m is an arbitrarily chosen positive integer. I express this property of the series as follows: ‘the series has a determined limit b’. Hence these words serve only to enunciate this property of the series, without for the moment alluding to another; and as we connect the series to a particular sign b, we must in the same way give different signs b, b´, b" to different series of the same species.1
对于康托尔，实数与表示分数序列的符号一致，这些分数序列在接近极限时变得无限密集，并且在索引（表示大数的更简单方法）大于值n 1的情况下，其中暂时不需要实际可计算性，两个分数之间的绝对值差小于分配的任何正有理数ε 。从这个解释中可以看出，康托尔引入了正是由这些序列组成的实数时，本质上是形式主义的精神。符号 b表示序列，因此也是一个数字（理性或非理性），尽管它只是为了以简洁的方式表示系列而引入的一个符号。
For Cantor, the real numbers coincide with the symbols that denote sequences of fractions that become infinitely denser in proximity to the limit, and are such that in the case of indices (a simpler way of expressing large numbers) greater than a value n1, of which for the moment the actual calculability is not required, the difference in absolute value between two fractions is less than any positive rational number ε assigned. Discernible in this explanation is the essentially formalist spirit with which Cantor introduces the real numbers that consist precisely of these sequences. The sign b denotes the sequence and consequently also a number (rational or irrational), though it is only a symbol introduced to represent the series in a concise manner.
因此，根据定义，实数是基本序列或连续，它们的算术是相同的普通运算的形式扩展序列。对角数和横向数之间的关系序列是康托尔术语中的基础，它标识了用符号指定的实数
Hence the real numbers are, by definition, fundamental sequences or successions, and their arithmetic is a formal extension of the ordinary operations of the same sequences. The sequence of relations between diagonal and lateral numbers is fundamental in Cantor’s terms and identifies the real number designated with the symbol 2. The algorithm of approximation of 2 devised by the Pythagoreans was therefore a kind of effective realization of the non-extended point corresponding to the 2.
现在康托尔证明了任何不为空且不重叠的实直线区间的集合是一个至多可数的集合，也就是说，与正整数的集合一一对应. 相反，实数的聚集被设想为长度逐渐趋于 0 的区间序列，形成了一个不可数的集合。康托尔用著名的对角线论证证明了这一点。
Now Cantor demonstrated that any collection of intervals of the real straight line that are not null and do not overlap is a set that is at most numerable, that is to say, is in one-to-one correspondence with the set of positive whole numbers. Instead the agglomeration of real numbers conceived as sequences of intervals of a length tending progressively towards 0 forms a set that is not numerable. Cantor showed this with the famous diagonal argument.
康托尔的理论不是基于一条直线的无限可分性，该直线最终趋向于没有延伸的点，而是基于实际预先存在的不可数的无限点/数被认为是连续的区间，其长度逐渐减小并趋向于 0。如果我们正式地将一个符号与这些连续中的每一个联系起来（如康托尔提出的，指的是基本序列），我们可以将这些单个符号的延续视为原子点的集合在这种情况下，我们可以定义正长度的区间。最终，我们正在处理数学模型的符号构造虽然允许我们建立分析，但并不完全符合我们的直觉。但是现在两个极值a和b之间的实数集，其中a < b或区间 [ a , b ] 的长度由实数b – a精确定义，不等于 0。因此康托尔和戴德金的理论，如果它是连贯，实际上使我们从芝诺悖论中解脱出来，因为它允许我们定义长度有限且不为零的区间，并且由无数非扩展点的集合形成。
Cantor’s theory is not based on the indefinite divisibility of a straight line that tends, in the end, to points without extension, but on the actual pre-existence of a non-numerable infinity of points/numbers to be thought of as successions of intervals, the length of which progressively diminishes and tends towards 0. If we formally associate a symbol with each one of these successions (as Cantor proposed, referring to fundamental sequences) we can think of the continuation of these individual symbols as an aggregate of atomistic points in the context of which we can define intervals of positive length. Ultimately, we are thus dealing with a symbolic construction, of a mathematical model of the continuum that, while allowing us to build an analysis, does not wholly correspond to our intuition. But now the set of real numbers between the two extremes a and b, where a < b, or the interval [a, b ], has a length defined precisely by the real number b – a, which is not equal to 0. Hence the theory of Cantor and of Dedekind, if it is coherent, virtually extricates us from Zeno’s paradox because it allows us to define intervals of a length which is finite and not null and formed by an innumerable collection of non-extended points.
这样，如果我们相信怀特海的论点，即现实性具有严格的原子性特征，我们就可以成功地赋予连续统以现实性和功效的内涵。此外，数学是理解这种原子实在论的思想是否能够产生可接受的和连贯的理论的基本工具。正如约翰洛克所指出的（关于人类理解的论文, 二, 17, 标准杆。9），我们在数字中找到了最充分的概念，可以理解无限的本质，并掌握无限将我们拖入其中的元素的混乱积累，并且没有其他充分的手段，心灵最终会落入其中通过迷失自己。物理学也研究原子。但物理学和数学在这个领域追求不同的目标：如果物理学家研究思想与现实之间的相互关系，数学家则寻求澄清思想与公式之间的联系，从而倾向于阐述适当且足够合理的问题。我们的感知模型，这是将它们应用于现实世界的必要先决条件。然而，尽管有这些不同的探究方向，
In this way we can succeed in giving to the continuum a connotation of actuality and efficacy, if we are to credit Whitehead’s thesis that actuality has a strictly atomistic character. Mathematics, moreover, is the essential tool for understanding whether this idea of atomistic realism can result in an acceptable and coherent theory. As John Locke noted (An Essay Concerning Human Understanding, II, 17, par. 9), we find in numbers the most adequate concept for understanding the nature of the infinite, and for getting to grips with that confused accumulation of elements into which the infinite drags us, and within which the mind, without other sufficient means, ends up by losing itself. Physics also studies atoms. But physics and mathematics pursue different aims in this field: if the physicist studies the reciprocal relation between thought and reality, the mathematician instead seeks to clarify the nexus between thought and formulas, and as a result is oriented towards the elaboration of appropriate and sufficiently plausible models of our perceptions, a necessary prerequisite for them to be applied to the real world. Despite these differing orientations of inquiry, however, both physics and mathematics seek to establish the reality of the world.
然而，我们留下了与这个问题有关的各种问题。实数是抽象域的一个元素，其公理由各种可能不是数字的数学实体所满足。因此，我们只能通过同构来寻求定义数学实体的本质，而缺乏分类性几乎是不可能消除的。2此外，我们被迫把数轴上的一点想象成一个不可分割的实体。那么我们能否将这个数字概念改编为一个基本序列、一个部分，或者更确切地说是一个区间序列？Dedekind 和 Cantor 都觉得有必要以某种方式创造一个新的实体，它对应于一个序列或一个部分，并且可以用一个符号来指定，我们可以给它分配与任何整数一样多的现实。3
We are left, however, with various problems relating to this matter. A real number is an element of an abstract domain the axioms of which are satisfied by a variety of mathematical entities that may not be numbers. We can therefore seek to define the essence of mathematical entities only by means of an isomorphism, with a lack of categoricalness almost impossible to eliminate.2 Moreover, we are compelled to think of a point on the number line as an indivisible entity. Can we then adapt this concept of number as a fundamental sequence, as a section or rather as a sequence of intervals? Both Dedekind and Cantor felt obliged to create in some way a new entity that would correspond to a sequence or to a section, and that could be designated with a symbol to which we could assign as much reality as to any whole number.3
对于超限数（大于所有有限数但不一定绝对无限），康托尔基于两者的正式定义确定了类似于实数的本体论状态。那么，他应该在不同的场合坚持固有的现实，这并非巧合任何类型的数字，因为这些数字，无论多么抽象，都是原子类型的现实的基础——就一个完整和有序的数字领域的公理而言，唯一一个似乎有助于并列的数字，以连续体的迭代划分的唯一潜在无限。出于这个原因，不仅因为有必要以连贯的方式定义超限数，康托尔倾向于认为实际的无限在数学中是可以实现的。
For transfinite numbers ( larger than all finite numbers but not necessarily absolutely infinite) Cantor fixed an ontological status akin to that of real numbers, on the basis of a formal definition of both. It is no coincidence, then, that he should insist on different occasions on the inherent reality of numbers of whatever kind, because these numbers, however abstract, were the basis of an actuality of an atomistic kind – the only one that seemed to lend itself to being juxtaposed, with respect to the axioms of a complete and ordered numerical field, to the merely potential infinity of an iterated division of the continuum. For this reason also, and not only because it was necessary to define transfinite numbers in a coherent way, Cantor was inclined to consider the actual infinite to be realizable in mathematics.
康托尔在宣布他的发现时附有声明：“以这种方式获得的新数字总是与前面的数字具有相同的具体精度和相同的客观现实 ”。4对于康托尔来说，整数已经是实际的实体，因为根据它们的定义，它们在我们的思想中具有完全确定和独特的位置，在某种程度上可以确定和修改其发展。数字的现实性意味着它们“必须被视为与智力不同的外部世界过程和关系的表达或图像 [ Abbild ]”。5
Cantor accompanied the announcement of his discoveries with the declaration that ‘the new numbers obtained in this way always have the same concrete precision and the same objective reality as the preceding ones ’.4 Whole numbers for Cantor are already actual entities, in the sense that, on the basis of their definition, they have a position which is perfectly determined and distinct in our thought, to the extent of determining and modifying its development. The actuality of numbers means that they ‘must be considered as an expression or image [ Abbild ] of processes and relations in the external world, distinct from the intellect’.5
这可以追溯到 1883 年单独出版的专着《多重性一般理论的基本原理》，这是康托尔关于实际无限部分的第一次公开倡导，它反对源自亚里士多德的悠久传统，除了少数例外——其中一个最重要的是伯恩哈德·博尔扎诺（Bernhard Bolzano）在 1851 年的《无限悖论》 ——赋予了无限仅仅潜在的意义。在Fundamentals中，康托尔对reelen Zahlen进行了区分，实数区别于复数（例如
This dates back to a monograph published separately in 1883, Fundamentals of a General Theory of Multiplicities, the first public advocacy on Cantor’s part of the actual infinite, in opposition to a long tradition originating with Aristotle that, with a few exceptions – one of the most important being Bernhard Bolzano’s Paradoxes of the Infinite in 1851 – had assigned to the infinite a merely potential significance. In the Fundamentals, Cantor made a distinction between reelen Zahlen, real numbers that are distinguished from complex ones (such as −1) in a mathematical-formal sense, and the realen Zahlen – numbers that have a real existence rather than a purely formal one, even when introduced in a purely formal way. The transfinite numbers, just like the irrational ones defined by the fundamental sequences, consequently had to possess a wholly analogous ontological status; the irrational in relation above all with points in geometrical space, the transfinite in relation to a material reality consisting of a combination of monads of (still) uncertain cardinality. The transfinite numbers were to be considered, Cantor noted, as new irrational numbers. Their mathematical definition, in formal terms, placed them on the same, identical level. Irrational and transfinite numbers resembled each other in their intrinsic nature, both were forms or variations of the actual infinite, and if we did not accept the one then we were forced, for consistency’s sake, not to accept the other.6 Defining the numerical atoms of the continuous line as fundamental sequences, of an infinite kind, of relations between whole numbers, Cantor thus inherited the atomistic realism of the Pythagoreans. The arithmós, the positive whole number, now seemed capable of founding a mathematics of the indivisibles, of the number-points of the continuum.
Gottlob Frege 为 Cantor 的论文辩护，比较以逻辑连贯性为名的各种数的本体论状态：
Gottlob Frege defended Cantor’s thesis, comparing the ontological status of numbers of various kinds in the name of logical coherence:
我发现自己完全同意他判断那些在所有可能的数字中只愿意承认有限自然数是实际的人的意见……在我们的研究中，我们可以毫无保留地使用任何名称或符号以逻辑上无可挑剔的方法引入；所以 ∞ 1最终与数字 1 和 2 一样合理。7
I find myself completely in agreement with him in judging the opinion of those who, among all possible numbers, would like to recognize only finite natural numbers as actual … In our research we may use, without the least reservation, whatever name or sign that has been introduced with a logically impeccable method; so that the number ∞1 ends up being as justified as the numbers 1 and 2.7
对于弗雷格来说，所有以逻辑连贯的方式引入的数字都是实际的。这是对现实的保证，有理数和无理数现在为连续体和所有分析理论提供了可能，这些理论可以确定某些数学实体，例如两条线之间的交点，实际上存在于连续体中. 这一立场推翻了迄今为止维持的整个时空理论，并驳斥了至少自莱布尼茨以来哲学家中最普遍的信念，即由点组成的空间在逻辑上是不可能的（罗素，原理，par . 423). 这一理论发展的关键时刻是康德对纯粹理性的批判，特别是其中讨论的关于物质组成的第二个自相矛盾（或两个相互矛盾的断言的共存）。二律背反的论点是肯定“每一个实体都是由简单的部分组成的，除了简单的部分和由它组成的东西之外，别无他物。” 8罗素能够解释说，由于最近的连续统算术理论，相反，对立的肯定不再可持续。他指出，反对该论点和反对点存在的论点（《原则》，第 435 段）将表明，无论是理性的还是非理性的，在康托尔和戴德金的观点中，数字都不应该与直线上的点相关联。在 1837 年的《科学学说》 (315, 7) 中，伯恩哈德·博尔扎诺 (Bernhard Bolzano) 已经直面这个问题，他认为康德的对立（即没有复合实体由简单的部分组成）的论证是完全错误的。博尔扎诺将连续统设想为简单且不可分割的实体的集合，例如点或瞬间。总的来说，他的理论是不充分的：在缺乏完整性标准的情况下，理性语料库的空白仍然存在。尽管如此，博尔扎诺（Bolzano）在他死后发表的《无穷悖论》（Paradoxes of the Infinite，1951 年出版）中预见到了数字原子论，它将继续表征本世纪末出现的更严格的算术连续统理论，
For Frege, all numbers that were introduced in a logically coherent way were actual. This was a guarantee of reality that the rational and irrational numbers now provided for the continuum and for all the theories of analysis that made it possible to establish that certain mathematical entities, such as the point of intersection between two lines, actually exist in the continuum. This was a position which overturned the whole theory of space-time maintained up to this date, and refuted the most widespread belief among philosophers, at least since Leibniz, that a space made up of points is not logically possible (Russell, Principles, par. 423). A crucial moment in the development of this theory had been Kant’s Critique of Pure Reason, in particular the second antinomy (or coexistence of two contradictory assertions that may both be justified) discussed therein, on the composition of substances. The thesis of the antinomy consisted of the affirmation that ‘every substance that is composed consists of simple parts, and nowhere does there exist anything other than the simple, and that which is composed of it.’8 Russell was able to explain that, thanks to recent arithmetic theories of the continuum, the contrary, antithetical affirmation was no longer sustainable. The argument used against the thesis and against the existence of points, he noted (Principles, par. 435), would show that neither should numbers exist, rational and irrational, that in Cantor and Dedekind were associated with points on a straight line. In the Doctrine of Science (315, 7) of 1837, Bernhard Bolzano had already confronted the question, arguing that the demonstration of the Kantian antithesis, according to which no composite substance consists of simple parts, was completely mistaken. Bolzano conceived of the continuum as an aggregate of simple and indivisible entities, such as points or instants. Taken as a whole, his theory was inadequate: in the absence of a criterion of completeness, the lacunae of the rational corpus remained. Nevertheless, Bolzano anticipated – including in his Paradoxes of the Infinite, published posthumously in 1951 – the numerical atomism that would go on to characterize the more rigorous theories of the arithmetical continuum that appeared towards the end of the century,
亨利柏格森意识到现实主义的要求，这种现实主义既实用又哲学，隐含在空间和时间的点状组成部分中。柏格森惊讶地意识到，生命这一对哲学如此重要的问题，迄今为止被数学完全忽视了。9对数学家来说，时间和空间被放在同一水平线上，就好像它们是同一类事物，而从一个到另一个所需要的只是用“继承”交换“并列”。柏格森指出，当我们的智能必须处理以某种方式固定的点时，我们的智能会更轻松地运行：它会问自己一个移动的东西在某个瞬间找到自己，在接下来的瞬间它会在哪里，以及经过哪里它将通过; 然而，即使它似乎对时间的持续时间如此感兴趣，“它总是想处理不动性，无论是真实的还是潜在的”。10但是在柏格森看来，数学连续统的意义和范围被颠覆了：正是由于真实或现实的事实，为了最终回应我们的实际需求，数学时空背叛了生活经验和内在时间。然后是诺伯特·维纳（Norbert Wiener），在与埃伯哈德·霍普夫（Eberhard Hopf – 还用于解释恒星的物理特性和原子弹的功能。与 Hopf 一起，Wiener 引入了一类积分方程（称为 Wiener-Hopf 方程），该方程将继续用于模拟最多样化的时间过程，并采用有效的信号预测和滤波技术。同样在维纳的活动和研究的背景下，随着基于反馈概念的设备和数学模型的研究，出现了与普鲁斯特和柏格森的愿景一致的方向。控制论的时代在某些方面不同于物理学家的时代，它也必须是目的论现象的时代，是成长和学习过程的时代——它与真实经历的持续时间非常相似，内部时间被组织为意识状态的相互（相互）渗透的过程。它的结构，基于现象体现，可以用时间过程的数学模型来表达，12用积分和 Toeplitz 矩阵表示，它表达了与古代数学相同的数字增长的想法——通过连续的日光校正。
Henri Bergson was aware of the demands of a realism that was both practical and philosophical that was implicit in punctiform components of space and time. Bergson had been struck by the realization that lived time, a matter so central to philosophy, had thus far been completely ignored by mathematics.9 Time and space, for mathematicians, were placed on the same level, as if they were things of the same kind, and all that was needed to pass from one to the other was to exchange ‘succession’ for ‘juxtaposition’. Our intelligence, Bergson noted, operates with more ease when it has to deal with points that are in some way fixed: it asks itself where a moving thing finds itself in a certain instant, where it will be in a subsequent instant, and through where it will pass; yet even if it appears to be so interested in temporal duration ‘it always wants to deal with immobility, real or potential’.10 But with Bergson the meaning and the scope of the mathematical continuum are overturned: precisely due to the fact of being real or actual, in order in the end to respond to our practical needs, mathematical time and space betray lived experience and interior time. It was then Norbert Wiener, during the course of his collaboration with Eberhard Hopf, who realized that Bergsonian time – not a linear flow but an ‘emboîtement or nesting of mental events one within another, in the gradual enrichment of the I’11 – served also to explain the physics of stars and the functioning of the atomic bomb. Together with Hopf, Wiener introduced a class of integral equations (known as Wiener–Hopf equations) that would go on to be used to simulate temporal processes of the most varied kind, with efficient techniques of signal prediction and filtering. Also in the background of Wiener’s activities and research, with the study of devices and mathematical models based on the concept of feedback, an orientation emerged that was in keeping with the vision of Proust and of Bergson. The time of cybernetics, which differed in some ways from that of physicists, had also to be that of teleological phenomena, of processes of growth and learning – and so resembled duration as it was really experienced, as interior time organized as a process of reciprocal (inter)penetration of states of consciousness. Its structure, based on the phenomenon of emboîtement, could be expressed in the mathematical models of temporal processes,12 in terms of integrals and Toeplitz matrices, which expressed the same idea of the growth of figures – by way of successive gnomonic corrections – as in the mathematics of antiquity.
Toeplitz 矩阵的形式（这里只写了六行六列）的特点是平行于主对角线的线上的所有元素相等，因此第一行和第一列足以定义它。嵌入现象在矩阵内部可以分割出完全相同的 Toeplitz 形式的较小矩阵这一事实是显而易见的。这种自相似性（与自身的一部分完全或近似相似的对象）的这种特性让人想起希腊几何中的类似结构，在这种情况下反映在逆矩阵的结构中，13它可以从第一个自生成行和列。
The form of Toeplitz matrices ( here written for only six rows and six columns) is characterized by the equality of all the elements on the lines that are parallel to the principal diagonal, hence the first row and the first column are sufficient to define it. The phenomenon of emboîtement is visible in the fact that inside the matrix it is possible to segment smaller matrices of exactly the same Toeplitz form. This property of self-similarity (of an object that is exactly or approximately similar to a part of itself ) recalls similar constructions in Greek geometry, reflected in this case in the structure of the inverse matrix,13 which can self-generate from the first row and column.
Toeplitz 矩阵是 Wiener-Hopf 方程离散化（将连续函数转换为离散对应项）的直接结果，多年来在矩阵代数中获得了极大的重要性，将其自身作为结构的代数概念的原型参考点的一个矩阵。在这样的结构中，人们必须寻找计算效率的原因，以及生存时间和数学时间之间密切相关的原因。
Toeplitz matrices, the direct consequence of the discretization (transforming continuous functions into discrete counterparts) of Wiener–Hopf equations, would acquire over the years a great importance in matrix algebra, presenting themselves as an archetypal point of reference for the algebraic concept of the structure of a matrix. In such a structure one had to look for the reasons for computational efficiency, but also the reasons for the affinity between lived time and mathematical time.
因此，实数的算术连续统为同样的应用数学提供了基础，而数字、有序形式对应于物理或生物的精确模型现实，可以通过类比来回忆生活时间的现实，即柏格森将真实的主观时间与数学家的空间化时间进行对比的现实。维纳认为，这取决于可以随意假设物理、化学、生理或数学意义的单一和特定过程。
Consequently the arithmetical continuum of real numbers provided the foundation for the very same applied mathematics, and the numbers, ordered forms corresponding to precise models of physical or biological reality, could recall by analogy the reality of lived time, of the authentic subjective time that Bergson had contrasted to the spatialized time of the mathematicians. This depended, Wiener believed, on single and specific processes that can assume at will a physical, chemical, physiological or mathematical significance.
尽管如此，很快就发现，首先要感谢 Brouwer，在算术连续统的新理论中，用于定义实数的基本级数不响应计算有效性的标准。事实证明，这是一个至关重要的发现，与数学基础的危机有关，在此过程中，现实性和有效性将成为 20 世纪数学家将承担定义任务的另一个概念的特权——算法。
Notwithstanding this, it was soon discovered, thanks above all to Brouwer, that the fundamental series that served to define real numbers, in the new theory of the arithmetical continuum, do not respond to a criterion of computational effectiveness. This proved to be a crucial finding, tied to the crisis in the fundamentals of mathematics, in the course of which actuality and effectiveness would become the prerogative of another concept that the mathematicians of the twentieth century would assume the task of defining – the concept of algorithm.
12. 数字的真实性：戴德金的部分
12. The Reality of Numbers: Dedekind’s Sections
逻辑在数学发展中的重要性是公认的。但可以说数学源于逻辑吗？这个问题不同于关于数学可还原为逻辑的问题，在 19 世纪末和 20 世纪上半叶之间，理查德·戴德金、戈特洛布·弗雷格和伯特兰·罗素试图给予肯定的回答。
The importance of logic in the development of mathematics is universally acknowledged. But can it be said that mathematics derives from logic? The question is different from the one regarding the reducibility of mathematics to logic, which between the end of the nineteenth and the first part of the twentieth centuries Richard Dedekind, Gottlob Frege and Bertrand Russell had sought to answer in the affirmative.
如果我们看看古代建立数学和逻辑学科的原则，我们仍然陷入两难境地：三段论所依据的推理是否有其独立的起源和证明，或者它是从数学中诞生的？ ? 相对较新的调查至少证明了第二个假设的合理性：对亚里士多德的三段论理论的仔细解释（先验分析，25 b 26 ff.）已经可以识别一种推理和相应的术语选择带我们回到希腊的比例理论。1
If we look at the principles that in antiquity founded the disciplines of both mathematics and logic, we remain caught in a dilemma: does the reasoning on which the syllogism is based have its own independent origin and justification, or is it born out of mathematics instead? Relatively recent investigations have demonstrated at least the plausibility of the second hypothesis: a careful interpretation of the syllogistic theory of Aristotle (Prior Analytics, 25 b 26 ff.) already makes it possible to recognize a kind of reasoning and a corresponding selection of terms that take us back to the Greek theory of proportions.1
反过来，关于比例理论又能说些什么呢？我们是在处理数学理论，还是逻辑理论，经过深思熟虑并穿着数学术语？这个问题不是一个似是而非的问题，因为欧几里得的比例理论可以伪装成逻辑（汉斯·赖兴巴赫在 1928 年就已经隐含地争论了这一点）2——而且因为欧几里得《元素 》第五卷中的比例定义在十九世纪末，对数概念显然具有逻辑性质的延伸。
What can be said, in turn, about the theory of proportions? Are we dealing with a theory of mathematics, or with a theory of logic, thought through with, and dressed up in, mathematical terms? The question is not a specious one, because the Euclidean theory of proportions could pass for logic in disguise (something implicitly contended already in 1928, by Hans Reichenbach)2 – and because the definition of proportion in Book V of Euclid’s Elements gave rise, at the end of the nineteenth century, to an extension of the apparently logical nature of the concept of number.
正如在横数和对角数的例子中已经很明显的那样，一个实数可以由无限序列的有理极值区间组成，每一个区间都包含在前一个区间内，其长度逐渐趋向于 0。命题，假设存在一个包含在所有区间中的点。此外，给定区间序列，我们可以找到一个将有理数语料库划分为两类的标准，使得第一类的每个数字都劣于属于第二类的任何数字。3
As is already evident in the example of lateral and diagonal numbers, a real number can consist of an infinite sequence of intervals with rational extremes, each one contained within the preceding one, the length of which tends progressively towards 0. On the basis of a proposition, one assumes that a point exists that is contained in all of the intervals. Besides, given the sequence of intervals, we can find a criterion for dividing the corpus of the rational numbers into two classes, such that every number of the first class is inferior to any number belonging to the second.3
图 7
Figure 7
基于这个图式，算术连续统的模型将在 19 世纪末被详细阐述，这些模型基本上等同于康托尔的基本序列的模型。Dedekind，特别是4，会详尽地解释，每当我们有一个标准将每个有理数分配给两类有理数A和B中的一类且仅一类时，就会定义一个无理数x，其中每一个数A不如B的所有数量，而且，其中A不具有最大元素，B不具有最小元素。5在 Dedekind 的语言中，对 ( A , B ) 定义了有理数语料库的一个部分( Abschnitt )。现在Eudoxos/Euclid的比例理论具有相同的含义。它确定，给定四个量a、b、c和d ，如果a : b是，则a : b和c : d两个关系相等每次c : d分别大于、等于或小于m : n时，大于、等于或小于关系m : n（m和n整数） 。6在这种情况下，关系a : b将有理数集分为两类A和B，第一类由所有低于a : b的关系m : n形成，第二类由所有关系m 形成：n优于a : b。以类似的方式，可以根据关系c : d定义两个类C和D，然后我们可以证明A = C和B = D。换句话说，如果关系a : b和c : d在欧几里得的意义上是相等的（Elements , V, Def. 5），那么它们标识相同的部分，也就是说，定义这些部分的类的相对对是共广的。
Based on this schema, models of the arithmetical continuum would be elaborated towards the end of the nineteenth century that were substantially equivalent to that of Cantor’s fundamental sequences. Dedekind,4 in particular, would explain to an exhaustive degree that an irrational number x is defined every time that we have a criterion for assigning every rational number to one and only one of two classes of rational numbers A and B, where every number of A is inferior to every number of B and, moreover, where A does not possess a maximum element and B does not possess a minimum one.5 In the language of Dedekind, the pair (A, B ) defines a section (Abschnitt ) of the corpus of rational numbers. Now the theory of proportions of Eudoxos/Euclid has the same meaning. It establishes that, given four quantities a, b, c and d, the two relations a :b and c :d are equal if a :b is greater than, equal to or less than the relation m :n (m and n whole numbers) every time that c :d is, respectively, larger than, equal to or less than m :n.6 In such a case the relation a :b divides the set of the rational numbers into two classes A and B, the first formed by all the relations m :n that are inferior to a :b and the second by all the relations m :n superior to a :b. In a similar way it is possible to define two classes C and D in correspondence to the relation c :d, and we can then demonstrate that A = C and B = D. In other words, if the relations a :b and c :d are equal in Euclid’s sense (Elements, V, Def. 5), then they identify the same section, that is to say, the relative pairs of classes that define the sections are co-extensive.
该部分的概念相对抽象，适合于对实数进行逻辑定义，例如罗素或奎因提出的定义；但是我们已经看到了如何从计算对角线数和横向数之间的关系d : l的算法推导出一个典型的截面示例
The idea of the section is relatively abstract and lends itself to a logical definition of real numbers such as that proposed by Russell or Quine; but we have seen how a typical example of a section is derived from the algorithm that calculates the relations d :l between diagonal and lateral numbers to approximate 2. Of course, this algorithm was one of the many possible ones, conceived of in remote eras, which might have given rise to the idea of Eudoxos/Euclid, and after them to that of Dedekind. But it has a paradigmatic aspect, because implicit within it are the principal concepts from which a definition of real numbers was extrapolated. The property of completeness of real numbers that expresses the absence of gaps can be formulated by asserting that every Dedekind section is generated by a real number. The property of completeness may be defined in many equivalent ways, and can be attributed to any domain K such that every one of its elements generates a section in K, though it is evident that its reason for being is founded on the existence of algorithms which can approximate irrational numbers. The completeness (every element generates a section) and the density (given in any case two distinct elements a and b, where a < b, there exists another element greater than a and less than b ) are the two essential properties for an ordered set to be said to be continuous. The rational numbers form a dense but not complete set; the whole numbers form a set which is not dense but is complete; the real numbers form a set that is both dense and complete.
但我们被提示问自己：有什么需要定义
But we are prompted to ask ourselves: what need is there to define 2 as a section of the rational corpus rather than as a number (of infinite decimal digits) that when squared gives 2 as a result? We do not exclude the possibility of expressing an irrational number as an (aperiodic) sequence of infinite digits, nor the possibility of defining an irrational number, in some specific cases, as the number that raised to a certain power gives a whole number as a result. But behind the concept of section there is above all the idea of the relation between two quantities (2 and π are the relations, respectively, between the diagonal and the side of a square, and between a circumference and its diameter – and Dedekind resorts explicitly to the Eudoxos/Euclid theory of relations), and to say that this relation corresponds to an irrational number means that the latter is not expressible as a relation between whole numbers (Elements, X, 7).
因此，要了解什么是无理数，我们首先要知道什么是关系，也就是说，用符号“：”表示的两个量之间的关系有什么意义。为了定义关系的概念，我们可以依赖两种理论：起源于c的 Eudoxos/Euclid 理论。350 BC ( Elements , V, Def. 5) 和 Dedekind 所指的，以及一个亚里士多德提到了相似的几何概念（主题，158 b 29 ff.）：
Hence, in order to understand what an irrational number is we must first know what a relation is, that is to say, what meaning can be attributed to the relation between two quantities that is expressed with the sign ‘:’. To define the concept of relation we may rely then on two theories: that of Eudoxos/Euclid, which originated in c. 350 BC (Elements, V, Def. 5) and to which Dedekind refers, and the one to which Aristotle alludes, referring to the geometrical idea of likeness (Topics, 158 b 29 ff.):
同样，在数学中，有些事情似乎因为缺少定义而不容易证明，例如，平行于边的直线切割平行四边形，类似地分割线和面积。但是，一旦定义了这个定义，该属性就会立即显现出来；适用于面积和线的倒数减法（运算）是相同的（或给出相同的结果）；这就是“相同比率”的定义。7
In mathematics, too, some things would seem to be not easily proved for want of a definition, e.g. that the straight line, parallel to the side, which cuts a parallelogram divides similarly both the line and the area. But, once this definition is stated, the said property is immediately manifest; for the (operation of) reciprocal subtraction applicable to both the areas and the lines is the same (or gives the same result); and this is the definition of the ‘same ratio’.7
现在，正如我们在正方形的对角线和边的例子中看到的那样，anatanaíresis由一个基本的测量操作过程组成，每次从另一个量减去一个量时，它都会自行解决。用另一个更小的y作为测量单位来测量一个大小x ，简单来说，正如 Bernhard Riemann 在 1854 年指出的那样， 8将一个量叠加到另一个量上并固定较小量y的q次到较大的x（q为商）。如果存在除 0 以外的余数r（小于y )，重复测量y和r的操作，也就是说我们计算r包含在y中的次数。在数量不可比的情况下，这个过程会无限重复，因为没有余数是零。这实际上意味着将幅度x除以幅度y。使用在过程过程中获得的商q，我们构造（稍后将阐明）定义对应于不可通约量x和y之间关系的无理数的连续分数.
Now, as we have seen in the case of the diagonal and the side of the square, the anatanaíresis consists of a process of elementary operations of measurement that resolves itself every time in a subtraction of one quantity from another. To measure a magnitude x with another y that is smaller, taken as a unit of measurement, means in simple terms, as Bernhard Riemann noted in 1854,8 superimposing one quantity over another and fixing the number q of times that the smaller quantity y goes into the larger x (q for the quotient). If there is a remainder r other than 0 (smaller than y ), the operation of measurement of y and r is repeated, which is to say that we count how many times r is contained within y. The process is repeated indefinitely in the case of incommensurable quantities, because no remainder is nil. This means, in effect, dividing the magnitude x by the magnitude y. With the quotients q obtained in the course of the process we construct, as will later be clarified, the continuous fraction that defines the irrational number corresponding to the relation between the incommensurable quantities x and y.
涉及截面概念的关键现象现在如下：通过我们逐步构建的antanaíresis，用数字q确定一个数量包含在另一个数量中的多少次，数字分数m : n，其值交替大于或小于关系x : y。9换句话说，antanaíresis用于定义数值分数的两个连续A和B ，分别递增和递减，其中A中的每个分数都小于x之间的关系和y，并且B中的每个分数都更大。A中的分数是缺陷的近似值，B中的分数是超过相同比率x : y的近似值。10然后，我们可以通过赋予定义中涉及的数值比率m : n与用antanaíresis构造的数值比率相同的含义来破译欧几里得关系等式的定义。两个幅度之间的比率相等是根据这些比率与数字之间的比率所具有的顺序关系（>、<或=）来确定的；但是，如果我们想为这些比率赋予真实存在，我们需要将它们想象为通过antanaíresis或其他具有足够效率的程序的计算过程的结果。
The critical phenomenon that involves the concept of section is now the following: with the antanaíresis we construct, step by step, with the numbers q that establish how many times a quantity is contained within another, numerical fractions m :n, the value of which is alternately greater than or less than the relation x :y.9 In other words, the antanaíresis serves to define two successions A and B of numerical fractions, increasing and decreasing respectively, where every fraction in A is smaller than the relation between x and y, and every fraction in B is larger. The fractions in A are approximations by defect, those in B are approximations by excess of the same ratio x :y.10 We could then decipher the Euclidean definition of the equality of relations by giving to the numerical ratios m :n involved in the definition the same meaning as the numerical ratios constructed with antanaíresis. The equality of the two ratios between magnitudes is decided on the basis of the order relation (>, < or =) that such ratios have with the ratios between numbers; but if we want to assign a real existence to these ratios we need to imagine them as the result of a process of calculation by means of antanaíresis or of another procedure with a sufficient degree of efficiency.
当西蒙娜·威尔在给她哥哥安德烈的一封信中暗示，不可通约性的发现不仅令人痛苦，而且值得庆祝，我们需要考虑数字关系的可计算性：有可能“看到没有定义的东西”尽管如此，通过数字仍然始终是一种关系'，11这要归功于这样一个事实，即通过数字定义的缺失无论如何都会被计算过量和缺陷的数值近似值的程序所抵消。
When Simone Weil suggests in a letter to her brother André that the discovery of incommensurability was not only traumatic but also a cause for celebration, we need to think of the calculability of the numerical relations: it is possible ‘to see that what is not defined through numbers is nevertheless still always a relation’,11 thanks to the fact that the absence of a definition through numbers is in any case offset by procedures that calculate numerical approximations by excess and defect.
连续统无限的深渊当然不会被近似的程序所穷尽具有数值比率的量级——也就是说，它并没有被可以计算的一组数字所耗尽，正如图灵在 1936 年所证明的那样，它只是可数的；12但所有这些程序都允许接近ápeiron的深渊，这不会导致无法弥补的限制和统一性的损失。正如普罗提诺所说（Enneads, VI, 6, 3)，'如果你依赖无限而不投掷它，就像一张网，某种界限，它会从你的掌握中逃脱，你将找不到任何单一的东西，因为如果是这样的话你已经定义了它'。在俄耳甫斯和毕达哥拉斯的传统中，至少从原子论者和柏拉图到尼科马科斯，整数及其关系是构成要素，是这张网的真正结。
The abyss of the infinity of the continuum is certainly not exhausted by the procedures that approximate magnitudes with numerical ratios – it isn’t, that is to say, exhausted by the set of numbers that may be computed, which, as Turing demonstrated in 1936, is only numerable;12 but the entirety of those procedures permits an approximation to the abyss of the ápeiron that does not entail an irreparable loss of limit and unity. As Plotinus stated (Enneads, VI, 6, 3), ‘if you rely on the infinite without throwing over it, like a net, some kind of delimitation, it will escape from your grasp and you will find nothing that is unitary, because if that were the case you would already have defined it’. In the Orphic and Pythagorean tradition, from the atomists and Plato up to Nicomachus at least, whole numbers and their relations were the constitutive elements, the real knots of this net.
当 Russell 和 Whitehead 在Principia Mathematica (1910-13) 中试图用逻辑来定义数学，特别是用类来定义无理数时，他们首先考虑的是 Dedekind 的截面概念。后来，奎因或多或少做了同样的事情，13以《原理》中阐述的定义为模型 ——这是戴德金和欧几里得的理论特别适合用纯逻辑术语来表述实数理论的一个明确迹象. 罗素在《数学原理》（第 259 和 261 段）中详细解释了这一点。罗素的愿景预设了一个普遍和存在量词的理论，基于诸如all a等术语的使用，some a , every a , an a 存在，它将实数知识与命题函数的逻辑重新连接，并最终连接到可以从公理集合正式推导出的断言系统。此外，量词的逻辑通过该部分的数学思想和算法模式得到验证，而算法模式反过来又证明了它（第 60 段）。14
When Russell and Whitehead in Principia Mathematica (1910–13) tried to define mathematics in terms of logic, and in particular an irrational number in terms of classes, they were thinking above all of Dedekind’s concept of the section. Later on, Quine did more or less the same,13 taking as a model the definition elaborated in the Principia – a sure sign that the theories of Dedekind and Euclid lent themselves particularly well to the formulation of a theory of real numbers in purely logical terms. Russell explained this at length in the Principles of Mathematics ( pars. 259 and 261). Russell’s vision presupposes a theory of universal and existential quantifiers, based on the use of terms such as all a, some a, every a, an a exists, that reconnects the knowledge of real numbers to a logic of propositional functions, and definitively to a system of assertions that may be formally deduced from a collection of axioms. Furthermore, the logic of the quantifiers is validated by the mathematical idea of the section and by the algorithmic schema that in turn preceded and justified it ( par. 60).14
节的概念在多大程度上符合我们对数字的直观概念？Dedekind 的理论在某些方面是违反直觉的，因为将数字视为具有相对复杂结构（如截面）的集合似乎并不自然。出于这个原因，如果该部分没有确定有理数，Dedekind 认为有必要创建一个新实体，即无理数α，其中α是旨在表示A类和B类对的符号。因此，Dedekind 的创作与 Cantor 的创作有着密切的联系。他在 1888 年写给海因里希·韦伯的一封信中也提到了这一点：
To what extent does the concept of section correspond to our intuitive idea of numbers? Dedekind’s theory is in some respects counter-intuitive, because it does not seem natural to conceive of a number as a collection characterized by a relatively complex structure such as the section has. For this reason, in case the section did not identify a rational number, Dedekind considered it necessary to create a new entity, an irrational number α, where α is a symbol designed to denote the pair of classes A and B. Hence Dedekind’s creations have an affinity with Cantor’s. He noted as much in a letter of 1888 addressed to Heinrich Weber:
这是完全相同的问题，关于您声称无理数应该是节本身，而我更喜欢创建与节相对应并产生节的新事物（不同于节）。我们有权将这种创造的力量归功于我们自己，以这种方式处理所有数字更为合适。甚至有理数也会产生截面，但我当然不会试图将有理数与这些产生的截面识别。15
It is exactly the same question, with regard to which you claim that the irrational number should be none other than the section itself, while I prefer that something new is created (different from the section) that corresponds to the section and produces the section. We have the right to attribute this power of creation to ourselves, and it is much more appropriate to proceed in this way to tackle all numbers. Even rational numbers produce sections, but I will of course not seek to identify rational numbers with the sections that these produce.15
这里的重点在于产生（hervorbringen ），这是一种归因于数字内在属性的能力。数字产生了这些部分，这要归功于一种效力，一种使人们重新思考柏拉图的智者（ 247d）的论点的动态，从而使它们成为实际和有效的实体。数字的现实源于此，源于它们强加新思路的能力，这些思路依次预示了新概念。创造概念的自由似乎适应了严格的必要性，而每次我们遇到这种必要性时，现实就会显现出来。
The emphasis falls here on producing (hervorbringen ), a capacity attributed to the intrinsic properties of numbers. Numbers generate the sections thanks to a kind of potency, a dýnamis that makes one think again of the theses of Plato’s Sophist (247d) and which thereby qualifies them as actual and efficacious entities. The reality of numbers arises from this, from their ability to impose new lines of thought that prefigure new concepts in turn. The freedom to create concepts seems to adjust to a strict necessity, and reality manifests itself every time we encounter such a necessity.
戴德金的理论回答了完整性 ：似乎有证据表明它符合我们直观的连续性概念，但仍不符合渐进性标准。如前所述，完整性与密度一起表征真实事物的数值连续体，并且是没有间隙的技术同义词：从表示无理数的部分的数字域中排除，将引入间隙这在连续集中是不允许的。16
Dedekind’s theory answered the criterion of completeness : it seemed to appeal, with proof, to our intuitive concept of continuity, while still not corresponding to a criterion of graduality. Together with density, as previously mentioned, completeness characterizes the numerical continuum of real things and is a technical synonym for the absence of gaps: with the exclusion from the domain of numbers of sections that represent irrational numbers, there would be introduced gaps the presence of which cannot be allowed in a continuous set.16
同样对于康托尔来说，表示基本序列的符号是通过创造行为引入的——不是任意的，而是由于有理数语料库的扩展的必要性17 。康托尔解释说，集合中元素数量的概念具有直接的客观表示，并且“集合中元素的数量与数字本身之间的关系证明了数字的真实性，即使它是无限的”。18康托尔和戴德金用他们的数值连续统理论推翻了康德关于数学时间和空间感知的先验性质的论点：连续统的概念比空间或时间更基本，并且基本上建立在这个想法之上数量和等级。19更进一步的情况为新数字的真实性提供了可信度。如果我们放弃对数值场20的所谓阿基米德性质的必要澄清（即没有无限大或无限小的元素），戴德金的连续统理论就等同于康托尔的理论。断言一个数值域K是阿基米德，并且使得每个基本序列在K等价于断言在K中每个 Dedekind 部分都是由属于K的元素s生成的。21
For Cantor also, the sign that denotes the fundamental sequence was introduced by means of an act of creation – not of an arbitrary kind but due to the necessity17 for an extension of the corpus of rational numbers. The notion of a number of elements of a set has an immediate objective representation, Cantor explained, and ‘the relation between the number of elements of a set and a number itself demonstrates the reality of the number even when it is infinite’.18 With their theory of the numerical continuum Cantor and Dedekind overturned the Kantian thesis of the supposed a prioristic nature of the perception of time and space for mathematics: the concept of the continuum was more primary than either space or time and was basically founded on the idea of number and class.19 A further circumstance gave credibility to the reality of the new numbers. If we forgo a requisite clarification of the so-called Archimedean property of a numerical field20 (i.e. of having no infinitely large or infinitely small elements), Dedekind’s theory of the continuum becomes equivalent to Cantor’s. To assert that a numerical field K is Archimedean, and such that every fundamental sequence has a limit in K, is equivalent to asserting that in K every Dedekind section is generated by an element s belonging to K.21
在 19 世纪的最后几十年，一切似乎都集中在数字是真实的实体。在与康托尔的信件往来中，查尔斯·赫米特承认，整数似乎形成了一个存在于我们之外的现实世界，其必然性与我们的感官所感知的自然现实具有相同的必然性。对于康托尔来说，数字的现实——完整的、有理的、无理的和超限的——是建立在与圣经相同的真理之上的，以及基于具体证据的。后者似乎证明了对无限数的神圣知识：奥古斯丁在《上帝之城》中解释了它(XII, 19)，这对康托尔来说成为支持他关于无限现实的数学理论的支持。奥古斯丁宣布的数字和知识之间的对应关系在诗篇 146 篇 4-5 中的 litotes 中清晰可辨，在这段经文中唱出了“计算星星的数量/并给每个人一个名字”但谁是还有“他的智慧无数的主 [ tês sunéseos autoû ouk ésti arithmós ]”。根据奥古斯丁的说法，对于宇宙之主来说，数字的无限并非不可理解，“即使没有无限的数字”（上帝之城，XII，18）。
In the last decades of the nineteenth century, everything seems to converge on the assertion that numbers are real entities. During an exchange of letters with Cantor, Charles Hermite confessed that whole numbers seemed to form a world of realities that exists outside ourselves, with the same character of necessity as natural realities that offer themselves up to be apprehended by our senses. For Cantor the reality of numbers – whole, rational, irrational and transfinite numbers – was founded, as well as on concrete evidence, upon the same truths as the Scriptures. The latter seemed to testify to a divine knowledge of infinite numbers: Augustine explained it in City of God (XII, 19), which for Cantor became a support in favour of his mathematical theory of the actuality of the infinite. The correspondence between numbers and knowledge, proclaimed by Augustine, was legible by litotes in Psalm 146, 4–5, in the passage that is sung of Him who ‘calculates the number of the stars / and gives to each a name’ but who is also the Lord ‘whose intelligence is without number [tês sunéseos autoû ouk ésti arithmós ]’. For the Lord of the universe, according to Augustine, the infinity of numbers is not incomprehensible, ‘even though there is no number for the infinite number’ (City of God, XII, 18).
在 19 世纪末，同样的截面概念也在意大利数学家中流传，尤利塞·迪尼尤其将其作为他实函数理论的基础。他为自己设定了阐述连续统算术理论的任务，正如康托尔和戴德金德所做的那样，它独立于几何学和任何基于空间和时间感知的预设。他的理论在各方面都与章节相似，事实上 Dini 和 Dedekind 相互引用。对于 Dini 来说，识别无理数的两个类，一个是递减的，另一个是递增的，由以下数字构成
The same idea of the section, at the end of the nineteenth century, was also circulating among Italian mathematicians, and Ulisse Dini in particular made it the foundation of his theory of real functions. He set himself the task of elaborating an arithmetical theory of the continuum, exactly as Cantor and Dedekind had, that was independent of geometry and any presupposition based on perceptions of space and time. His theory is in every way similar to that of sections, and in fact Dini and Dedekind cited each other. For Dini also, the two classes that identify an irrational number, one of decreasing and the other of increasing order, are constituted by numbers that
无限期地相互接近，并以一个唯一且确定的幅度相互接近，该幅度的存在是先验已知的，但仅考虑对应于两类有理数的量可能永远无法达到，这标志着这些类的数量之间的界限; 因此，如果也为这个量分配了一个对应的数字，那么这个数字将不是有理数，而是有某种东西与它相对应……实际上是某种真实的东西，因为与它相对应的量具有真实和适当的存在，并且尽管它可能永远不会通过对应于有理数的单个量来达到，因此它仍然可以根据需要近似它。22
approach each other indefinitely and at a unique and determined magnitude the existence of which is known a priori, but which may never be reached with only consideration of quantities corresponding to rational numbers of the two classes, and which marks the limit between quantities of these classes; hence if to this quantity a corresponding number is also assigned, then this number will not be rational but something will correspond to it … effectively something real, given that a quantity corresponds to it that has a real and proper existence and that, although it may never be reached by way of the single quantities corresponding to rational numbers, is such that it may nevertheless approximate it as required.22
迪尼反复提到数字的真实存在，以表明它们不仅仅是虚构的，并且表示它们的符号，其唯一目的是尊重“语言的简单性”，23对应于实际和有形的现象，即使，我们可能会补充说，一个来历不明。
Dini repeatedly alludes to the real existence of numbers to indicate that they are not mere fictions, and that the symbols by which they are denoted, for the sole purpose of respecting a ‘simplicity of locution’,23 correspond to an actual and tangible phenomenon, even if, we might add, one of unknown origin.
然而，奇怪的事情发生了。数字的连贯性和真实存在是如此确定，以至于从它们中提取有限和无限集的数学逻辑的一般原则似乎是合理的。到 19 世纪末，戴德金德的数论的目标正是证明集合的逻辑理论是数的直观概念和算术运算的递归结构的先决条件。但这种逻辑却敢于脱离约定俗成的界限。在那里，它遇到了悖论的墙壁，数学家开始质疑类的存在。实在论产生了一种怀疑的唯名论，它依赖于逻辑语言与本质上是几个世纪以来已知并根据现代连续统理论完善的数学性质。
And yet something peculiar occurred. There was such certainty about the coherence and real existence of numbers that it seemed plausible to extract from them the general principles of a mathematical logic of finite and infinite sets. Towards the end of the nineteenth century Dedekind’s theory of numbers had precisely the aim of evidencing a logical theory of sets as a prerequisite of the intuitive notion of numbers and of the recursive structure of arithmetical operations. But this logic dared to depart from the consented limits. There it came up against the wall of paradoxes, and mathematicians began to call into question the existence of classes. Realism gave rise to a sort of sceptical nominalism that relied on the coherence of the language of logic with essentially mathematical properties that had been known for centuries and perfected according to modern theories of the continuum.
目的是寻求从逻辑原理中推导出一切，但数字的现实不太可能依赖于量化逻辑，该逻辑使用“ x存在使得f ( x ) = 0”或“每个x满足某个属性 P'——Russell 在处理 Dedekind 的部分时使用的那种。相反，将数学属性简化为命题函数的逻辑强加了一种唯名论的观点，它剥夺了现实的数量，将它们的每一个证据特征都转移到语言公式的准确性和形式的连贯性的问题上。系统。
The aim was to seek to derive everything from logical principles, but it is not likely that the reality of numbers depends on a logic of quantification that uses expressions of the type ‘x exists such that f (x ) = 0’, or ‘every x satisfies a certain property P’ – the kind Russell used when dealing with Dedekind’s sections. On the contrary, the reduction of mathematical properties to the logic of propositional functions has imposed a nominalistic vision which divested numbers of reality, shifting every one of their evidentiary features on to a matter of accuracy of linguistic formulas and the quality of coherence of a formal system.
罗素阐述了一种描述理论，目的是使语法主题从字面上消失，将它们简化为“不完整的符号”，根本不代表任何东西的符号。因此，有可能消除“鹰头马”或“方圆”存在的问题，否则它们就有可能被赋予普通语言使用中似是而非的对象的地位。阶级，以及阶级所定义的数字，也无法逃脱这种毁灭。表示类的符号，如描述中使用的符号，是不完整的符号。重要的是它们使用的正确性，但它们本身并没有任何意义：“因此，在我们介绍它们的方式中，类仅仅是语言或符号上的便利，而不是真实的对象……”24在一篇关于集合逻辑的论文中，奎因宣称，他的任务首先在于确定哪些命题是确定一个类的，或者，就哲学实在论而言，“存在哪些类”。25但从长远来看，每一种现实主义似乎危险的。实际上，对于奎因来说，类根本不存在，逻辑形式主义必须始终以消除暗示它们作为真实实体存在的所有显式用法为目标。因此，任何真正形式的知识最终都只是基于经验观察。
Russell elaborated a theory of descriptions with the aim of making grammatical subjects literally disappear, reducing them to ‘incomplete symbols’, symbols that do not represent anything at all. It thus became possible to exorcize the problem of the existence of the ‘hippogriff ’, or the ‘squared circle’, that otherwise risked being accorded the status of plausible objects in common language usage. Classes, and numbers defined by classes, do not escape either from this kind of annihilation. The symbols that denote classes, like those that are used in descriptions, are incomplete symbols. What matters is the correctness with which they are used, but in and of themselves they mean nothing: ‘hence classes, in the way we introduce them, are mere linguistic or symbolic conveniences, not authentic objects …’24 In a treatise on the logic of sets, Quine declared that his task consisted above all in deciding what the propositions are that identify a class or, in terms of a philosophical realism, ‘which classes exist’.25 But in the long run every kind of realism seems hazardous. In reality, for Quine, the classes do not exist at all, logical formalism must always have as its objective the elimination of every explicit usage that implies their existence as real entities. Consequently, any real form of knowledge ends up being based merely on empirical observation.
是什么导致了这种关于数字存在的分歧？当然，数学基础的危机——在 20 世纪初随着悖论的发现而达到顶峰——加剧了 Leopold Kronecker 已经表达的关于无限数存在的怀疑。康托尔滥用了数学家的抽象能力：这也是弗雷格的观点，根据弗雷格的观点，抽象有可能被用作几乎一种神奇的力量，而超限数构成了通过不可接受的魔术出现的实体。26自相矛盾的发现证明了那些数学家的判断是正确的，他们认为康托尔对无所不能的渴望是毫无根据的。
What caused this diffidence with regard to the existence of numbers? Certainly the crisis in the foundations of mathematics – which came to a head at the beginning of the twentieth century in the wake of the discovery of paradoxes – intensified the suspicion already expressed by Leopold Kronecker as to the existence of infinite numbers. Cantor had abused the mathematician’s power of abstraction: this was also the opinion of Frege, according to whom there was a risk that abstraction would be used as almost a magical power, and that transfinite numbers constituted entities made to appear by inadmissible conjuring tricks.26 The discovery of antinomies vindicated the judgement of those mathematicians who had attributed to Cantor a groundless aspiration to omnipotence.
然而，目睹戴德金德、康托尔和迪尼等数学家的断言与逻辑主义本身所引入的唯名论之间的粗暴转变仍然令人震惊。我们可以说，后者承担了使两个世纪以来提出的令人尴尬的问题变得无害的责任，尼采为此找到了最激进的表述：
It is nevertheless striking to witness the brusque transition between the assertions of mathematicians such as Dedekind, Cantor and Dini, and the nominalism that was ushered in by logicism itself. The latter, we might say, assumed the responsibility of rendering harmless the embarrassing question that had been posed for two centuries, and for which Nietzsche had found the most radical formulation:
问题仍然悬而未决：逻辑公理是否适用于真实，或者它们是创造真实的标准和手段，对我们来说是“真实”的概念？……然而，为了断言前者，正如已经说过的，有必要已经知道存在是什么——而这绝对不是案子。因此，该原则不包含真理的标准，而只是关于必须被认为是真实的东西的命令。27
the question remains open: are the axioms of logic adapted to the real, or are they criteria and means for creating the real, the concept of ‘reality’ for us? … In order to assert the former it is necessary, however, as has already been said, to know already what being is – and this is absolutely not the case. The principle therefore does not contain a criterion of truth but merely an imperative about that which MUST be taken to be true.27
此外，为了与从阿奎那到笛卡尔和威廉詹姆斯的悠久传统保持一致，正如怀特黑德和罗素在《数学原理》的导言中明确指出的那样，定义具有由选择最值得了解的事物强加的意志品质。然而，在这种意志行为的背后，有一种比尼采想要的更强烈的命令：那些不可改变和不可修改的现象的证据，它们具有相同的数字和相同的数学公式，是由最成熟和最有意义的人强加的。理论选择。
Moreover, in keeping with a long tradition stretching from Aquinas to Descartes and William James, and as Whitehead and Russell made clear in the introduction to Principia Mathematica, definitions have a willed quality imposed by the choice of that which appears most worthy of being known. And yet behind this act of volition there was an imperative even stronger than the one intended by Nietzsche: the evidence of those unalterable and unmodifiable phenomena, connatural with the same numbers and with the same mathematical formulas, that are imposed by the most established and meaningful theoretical choices.
13. 数学：发现还是发明？
13. Mathematics: A Discovery or an Invention?
数学是发现还是发明是一个经常被提出和滥用的问题，似乎总是需要一个批判性的回应，一个在真实与主观和任意之间的决定性区别。关于建立这种区分的可能性，数学和逻辑都注定要被称为见证。在确定公理、证明定理或构造方程的解时，思想似乎坚持真实和必然性的意志：一种交换，由于它们直接和原始的特性，试图区分两个术语在它们开始相互作用的精确时刻是对立的。但是他们最初的组合活动已经是一个无法解开的纠结，关于普锐斯是什么的问题，或先行者，真的是，介于意志和必然性之间，仍然没有答案。
Whether mathematics is a discovery or an invention is a question often posed and abused, seeming always to require a critical response, a conclusive distinction between that which is real and that which is subjective and arbitrary. Regarding the possibility of establishing such a distinction, both mathematics and logic are destined to be called to bear witness. In fixing axioms, in demonstrating theorems or in constructing the solution to an equation, thought seems to be adhering to the real, and the will to necessity: an exchange in which an attempt is made, due to their immediate and originary character, to distinguish two terms that are antithetical at the precise instant at which they begin to interact. But their initial combinatory activity is already an inextricable tangle, and the question concerning what the prius, or forerunner, really is, between will and necessity, remains unanswered.
引入实数的需要似乎与 Dedekind 提出的论点是对立的，即这是“人类思想的自由创造”，1但这种自由肯定必须符合与 Dedekind 所谓的规范一致的规范“思维法则”，即我们心灵将一件事与另一件事联系起来、使一件事与另一件事相对应、以一件事与另一件事代表的先天能力；没有它就不可能思考的能力。从我们出生的那一刻起，我们就进行这些连接和表征操作连续不断，但没有预定目标。结果，推理链似乎自然而然地排列起来，以便提出看似简单的概念，实际上是复杂的，例如数字的抽象概念，它通过连续的概括来表达自己，直到它达到实数和复数。Dedekind 指出，每一个代数和分析定理，无论多么先进，都可以用整数定理来表达，但这种情况绝不能妨碍新实体的创建：
The need to introduce real numbers seems to be antithetical to the thesis, advanced by Dedekind, that it is a matter of ‘free creations of the human mind’,1 but this freedom must surely correspond to a norm in keeping with those that Dedekind called ‘the laws of thought’, the congenital ability of our mind to connect one thing with another, to make one thing correspond to another, to represent one thing with another; an ability without which it would be impossible to think. From the moment of our birth we exercise these connective and representational operations continuously, but without a predetermined objective. The chains of reasoning seem as a result to arrange themselves, naturally, so as to suggest apparently simple notions, which are in fact complex, such as the abstract notion of number, which articulates itself through successive generalizations until it reaches real and complex numbers. Every algebraic and analytical theorem, Dedekind noted, however advanced, may be expressed with a theorem on whole numbers, but this circumstance must not impede the creation of new entities:
我看不出任何优点——狄利克雷也这么认为——真正完成这种令人厌烦的迂回说法，坚持使用和承认除了有理数之外什么都没有。相反，数学和其他科学中最伟大和最有利的进步总是由新概念的创造和引入组成，由于复杂现象的频繁重现而变得必要，而这些现象可能很难被旧概念控制。2
I can see no merit – and Dirichlet thought this too – in really fulfilling this wearisome circumlocution, and in using and recognizing with insistence nothing but rational numbers. On the contrary, the greatest and most advantageous progress in mathematics and in other sciences has invariably consisted of the creation and introduction of new concepts, rendered necessary by the frequent recurrence of complex phenomena that may only be controlled with difficulty by the old notions.2
庞加莱使用并无不同的论据来证明连续统的算术定义是正确的。因此，推动我们进行新创造的原因不仅仅取决于意志，而是取决于客观复杂现象的发生频率，这些现象可以方便地用新概念来标记。但是，允许自己被这些概念指定的东西中的隐含现实首先显示出自己具有产生它们的能力。通过这种方式，我们回到柏拉图智者（247 d-e）的一句话，根据该句子“实体只不过是生产的力量”，以及Śulvasūtra的数学，吠陀论文在其中主要问题，生产不断增加的数字序列。这个想法在 Dedekind 的工作中多次出现：数字在算术上下文中产生截面的方式与对应点在几何直线上产生截面的方式相同。3
Poincaré used not dissimilar arguments to justify an arithmetical definition of the continuum. Hence, the reason that pushes us to new creations does not depend on volition alone, but on the frequency of occurrence of objectively complex phenomena which it is convenient to label by way of new concepts. But the implicit reality in that which allows itself to be designated by these concepts reveals itself above all in the capacity to produce them. In this way we go back to a sentence of the Platonic Sophist (247 d–e), according to which ‘entities are nothing other than the power to produce’, and to the mathematics of the Śulvasūtra, the Vedic treatises that posed, among the principal problems, the production of sequences of figures of increasing scale. This idea recurs in the work of Dedekind on various occasions: the number produces the section in the arithmetical context in the same way in which a corresponding point produces it on the geometrical straight line.3
我们在 Karl Weierstrass 的数学中找到了关于生成有理数语料库部分的数的存在的相同论点。在 Salvatore Pincherle 收集的 Weierstrass 讲座的注释中，除其他外，它解释了我们如何在几何直线上建立数字和点之间的连接。如果您固定原点 O，以将数字链接到线 OB 的段，您可以通过类似于允许计算一个数字a使得 OB =一个OA，并且a，与 OB 相关的数，可以是有理数，也可以是无理数。反之亦然，要与由a测量的线 OB 的每个有限且确定的数字a相关联，必须将 B 视为两类点之间的分隔点，分别对应于更大和更少的数字，比一个。一个类似于 Dedekind 的想法，尽管没有明确地将实数定义为一个部分。在说明两个数（有理或无理）之间相等的概念时，魏尔斯特拉斯回收了一种推理形式，该形式等同于在 Eudoxos/Euclid 的理论中定义两个关系的相等性所需的推理形式（元素，五，定义。5），Dedekind 反过来会求助于定义数值连续性的概念。4
We find the same thesis on the existence of the number that generates the section of the corpus of rational numbers in the mathematics of Karl Weierstrass. In the annotations of Weierstrass’s lectures collected by Salvatore Pincherle, it is explained among other things in what way we may establish a connection between numbers and points on a geometrical straight line. If you fix the origin O, to link a number to the segment of the line OB, you measure OB as a function of the preassigned unit OA, by means of a procedure that is analogous to the antanaíresis or Euclidean algorithm that permits the calculation of a number a such that OB = aOA, and a, the number associated with OB, can be rational or irrational. Vice versa, to associate to every finite and determined number a a segment of the line OB measured by a, one will have to conceive B as the point of separation between two classes of points, corresponding to numbers that are greater and less, respectively, than a. An idea analogous to Dedekind’s, though not explicitly oriented towards the definition of a real number as a section. In the specification of the concept of equality between two numbers (rational or irrational) Weierstrass reclaims a form of reasoning equivalent to that which is required to define the equality of two relations in Eudoxos/Euclid’s theory (Elements, V, Def. 5), which Dedekind in turn would resort to in order to define the concept of numerical continuity.4
定义算术连续统的决定性是表示产生部分的数字的想法符号。Whitehead 指出，在符号的帮助下，我们能够在推理过程中执行几乎是机械的交易，否则需要执行最复杂的神经程序。也不应该养成每次做事都想一想自己在做什么的习惯。实际上，相反可能是可取的，因为进步通常是通过“扩展我们无需考虑就可以得出结论的重要操作的数量”来实现的。5
Decisive in defining the arithmetical continuum was the idea of denoting the numbers that produce sections with symbols. With the help of symbols, Whitehead noted, we are capable of carrying out transactions that are almost mechanical in the course of a reasoning that would otherwise require the most complex neural procedures to be undertaken. Nor should one cultivate the habit of thinking about what one is doing every time one does it. Actually, the reverse may be advisable, since advances often happen by ‘extending the number of important operations that we can conclude without thinking about it’.5
在他 1940 年的著名论文《数学家的道歉》中，戈弗雷·哈代毫不犹豫地以他自己定义为教条的方式直面数学现实的主题，其目的是避免任何误解。根据大多数数学家的说法，他声称相信一个数学现实不同于物理的，但观察到它的性质没有一致：有人认为它是精神的，我们是建造它的人；其他人则相信它需要一个独立于我们的外部世界。他指出，一个可接受的数学现实定义肯定有助于解决形而上学的问题，如果还包括对物理现实的解释，那么为什么所有问题都会得到解决。哈代表达了他的信念，就好像它们是一成不变的：
In his celebrated essay of 1940 entitled Apology of a Mathematician, Godfrey H. Hardy confronted the theme of the reality of mathematics without too many hesitations and in a manner that he himself defined as dogmatic, the aim of which was to avoid any misconception. He claimed to believe, in accordance with the majority of mathematicians, in a mathematical reality different from the physical one, but observed that there was no agreement as to its nature: some argue that it is mental, that we are the ones who construct it; others are convinced that it entails an external world that is independent of us. He pointed out that an acceptable definition of mathematical reality would certainly help to resolve the problems of metaphysics, and if an explanation of physical reality were also included, why then, all problems would be resolved. Hardy expressed his convictions as if they were set in stone:
我相信数学现实存在于我们之外，我们的任务是发现或观察它，并且我们证明的定理，夸耀它们作为我们的“创造”，只是对我们观察的注释。从柏拉图开始，这一观点得到了许多伟大哲学家的支持，而我使用的语言对分享它的人来说是自然的。6
I believe that mathematical reality exists outside of us, that our task is to discover or to observe it, and that the theorems that we demonstrate, qualifying them pompously as our ‘creations’, are simply annotations to our observations. This opinion, one way or another, has been supported by many great philosophers, from Plato onwards, and I use the language that is natural to one who shares it.6
无数的证据和证词证明，数学家创造的实体一旦将自己作为证据出现，就会打开一个未知的领域，每一个发现都在这里远不能与这些实体一起被视为自由意志的产物。但它似乎强化了这样一个事实，即第一个创造行为都不是脱离了一种内在的必然性，这种必然性迫使它被视为一种发现而不是一种发明。十九世纪末提出的实数定义是单一思想链的一部分，其背后的原因是基于一组与古代已经观察到的数字及其关系有关的属性，专门用于强加于数字这些特定定义而不是其他定义。在这些性质中，增长的主题被包括在内，而对古代数学家如此关注的数字放大的研究使我们能够瞥见将成为现代计算的核心主题的东西：
Countless proofs and testimonies attest that, no sooner have they presented themselves as evidence, than the entities created by a mathematician open up an unknown territory where every discovery is far from capable of being considered, along with those entities, as the product of free will. But it seems to reinforce the fact that neither is the first act of creation free from an intrinsic necessity that obliges it to be taken as more of a discovery than an invention. The definitions of real numbers proposed at the end of the nineteenth century are part of a single chain of ideas the reasons behind which are anchored in a set of properties pertaining to numbers and their relations already observed in ancient times, specifically designed to impose on numbers those particular definitions and not others. In these properties the theme of growth is included, and the study of the enlargement of figures that so preoccupied the mathematicians of antiquity allows us to glimpse what would go on to become the central subject of modern computation: the growth of numbers as a critical phenomenon for the stability of digital calculation.
14. 从连续体到数字化
14. From the Continuum to the Digital
Dedekind 解释说，如果空间是真实存在的，它不一定是连续的，因为即使它是不连续的，它的许多属性也会保持不变。1因此，空间的数学现实独立于无理数的存在：它们作为有理语料库的一部分的创造嵌入了一个已经形成和承认的现实，这显然是建立在整数及其关系之上的，这是解决无理数的真正解毒剂。无限的非存在，到ápeiron固有的结构的缺失。
Dedekind explained that if space has a real existence, it does not necessarily follow that it should be continuous, because many of its properties would remain the same even if it were discontinuous.1 The mathematical reality of space was consequently independent of the existence of irrational numbers: their creation as sections of the rational corpus embedded itself in a reality already formed and acknowledged, which was evidently founded on whole numbers and on their relations, the true antidote to the non-being of the infinite, to the absence of structure inherent in the ápeiron.
用 Dedekind 的话来说，人们可以从最顽固的建构主义数学家（例如 Leopold Kronecker 和 Henri Lebesgue）所支持的论点中找到一种正当理由，对他们来说，分析可以在没有数字集的情况下很好地完成，例如实数域数，比可数更强大。这是一个前提，它允许人们认为连续统一体与其说是离散的完成，不如说是对它的推断，本质上是不真实的，由模拟项目强加于我们日常经验中直观地出现的空间，和一个完整的，没有跳跃和没有间隙的事情。难道它没有受到我们对渐进式的不可抑制的需求的影响吗？，也就是说，对于一个被设想为“自由生成” 2而不是完整实体的集合的连续体？
In Dedekind’s words, one gleans a kind of justification ante litteram of the theses upheld by the most intransigent constructivist mathematicians, such as Leopold Kronecker and Henri Lebesgue, for whom the analysis could well do without the sets of numbers, such as the field of real numbers, of superior power to the numerable. This was a premise that allowed one to think of the continuum not so much as a completion of the discrete as an extrapolation of it, essentially unreal, imposed by a project of simulation of that which appears intuitively, in our everyday experience, as a space and a matter that are full, without jumps and without gaps. Was it not perhaps affected, that project, by our insuppressible demand for graduality, that is to say, for a continuum conceived as ‘free becoming’2 rather than as an assemblage of completed entities?
像戴德金德一样，克罗内克将无理数设想为有理数除法的产物语料库分为两类——也就是说，作为一个部分，但一个数字，如
Like Dedekind, Kronecker conceived of irrational numbers as the product of a division of the rational corpus into two classes – that is to say, as a section, but a number such as 2 was nothing other than a sign activated to denote this division. Henri Poincaré in Science and Hypothesis (1902) asked himself if in order to be sufficiently satisfied with a mathematical theory of the continuum, we should forget the origin of these signs. But could mathematical knowledge really base itself on such a manoeuvre?
我们习惯于将离散视为连续统的近似，在某种意义上，无理数近似于有理数的序列。但是为什么不颠倒这些术语并将连续统视为离散的近似呢？我们真正知道的是整数及其关系，我们称之为有理数——我们对现实的认识是基于这些数字，所有的计算都是有理的，也许没有必要将我们的知识扩展到数字，同时填补空白在理性语料库中，可能在人类或数字计算器的实际操作中并不是必不可少的。
We are used to thinking of the discrete as an approximation of the continuum, in the sense that irrational numbers approximate sequences of rational numbers. But why not reverse the terms and think of the continuum, on the contrary, as an approximation of the discrete? What we really know are whole numbers and their relations, which we call rational numbers – our knowledge of reality is based on these numbers, all calculations are rational, and perhaps there is no need to extend our knowledge to numbers that, while filling the gaps in the rational corpus, could turn out not to be essential in the actual operations of a human or digital calculator.
对于隐含在算术连续体中的原子论，我们在尼采晚期发现了一个激进的批评，尽管转移到了物理领域：
Of the atomism implicit in the arithmetical continuum we find a radical criticism in late Nietzsche, albeit shifted into the physical realm:
物理学家以他们的方式相信一个“现实世界”：一个对所有生物都固定的原子系统，具有必要的动力——因此对他们来说，“现象世界”被简化为这个方面，每个生物都可以进入它自己的世界方式，普遍存在和普遍必然性……但在这一点上，他们被欺骗了：他们假设的原子源自意识透视主义的逻辑，因此本身就是一种主观虚构。他们所描绘的这个世界形象与世界的主观形象并没有本质上的区别：它只是用更发达的感官，但它们始终是我们的感官……最终，它们在不知不觉中忽略了星座中的某些东西：实际上是必要的透视主义，每一个权力中心——不仅是人——都通过这种透视主义来构建其余的以自身为基础的世界，也就是说，根据它的光来衡量、建模和塑造它……他们忘记了将这种创造视角的力量考虑到“真实存在”中。3
Physicists believe, in their way, in a ‘real world’: an atomic system which is fixed for all beings, with necessary dynamics – so for them the ‘world of appearances’ is reduced to this aspect, accessible to every being in its own way, of universal being and universal necessity … But in this they are deceived: the atom that they postulate is derived from the logic of perspectivism of consciousness, and is consequently itself a subjective fiction. This image of the world that they draw is in no way essentially different from the subjective image of the world: it is just constructed with more developed senses, but they are always our senses … And in the end they have neglected something in the constellation without realizing it: in effect the necessary perspectivism by virtue of which every centre of power – and not only man – constructs the rest of the world on the basis of itself, that is to say, measures, models and shapes it according to its lights … They have forgotten to factor into ‘real being’ this force that creates perspectives.3
尼采提出的原则宣言是时代的标志，抓住了现代性的典型母题。这当然不是要破坏详细说明直觉连续统的数学模型的目的，即庞加莱在康托尔和戴德金的理论中有充分理由辨别出的那种组合。但至少它有助于抹黑这些理论，减少了建立称为部分、无理数、原子或点的东西在现实中存在的假装。. 康托尔、戴德金和庞加莱几乎将其保留为一种约束——迫切需要增加对“真实”尽管不可见世界的直接知识的感知（以及由更精确的仪器提供的放大器）——现在可以变成一种分裂或双重感知，在对现实的更激进主张与重申主观虚构性的抽象之间存在问题的张力。
The statement of principle launched by Nietzsche was a sign of the times and captured a typical motif of modernity. It was certainly not seeking to undermine the aim of elaborating a mathematical model of the intuitive continuum, of that kind of assemblage that Poincaré discerned with good reason in the theories of Cantor and Dedekind. But at least it contributed to discrediting those theories, cutting down to size the pretence of establishing that something called section, irrational number, atom or point exists in reality. That which Cantor, Dedekind and Poincaré retained as almost a constraint – the peremptory need to add to sentience (and to its amplifiers as furnished by instruments of ever greater accuracy) the direct knowledge of a ‘real’ albeit invisible world – could now change into a kind of scission or double perception, in a problematic tension between a more radical claim to actuality and an abstraction that reaffirmed subjective fictionality.
希尔伯特相信数学抽象和我们用来思考无限（包括算术连续统）的虚构必须通过“预先建立的和谐”在现实中建立基础。4但是现在数学本身出现了一个更引人注目的批评：集合论的悖论，从罗素在 1902 年传达给弗雷格的著名的二律背反论开始，使阶级的不分青红皂白的存在受到质疑，并在数学家中激发了对确定性的新基础和新基础的探索。
Hilbert believed that mathematical abstractions and the fictions with which we think about the infinite (including the arithmetical continuum) must have a foundation in reality by way of a ‘pre-established harmony’.4 But a more compelling criticism now emerged from mathematics itself: the paradoxes of the theory of sets, starting with the famous antinomy that Russell communicated to Frege in 1902, made questionable the indiscriminate existence of classes and activated, among mathematicians, the search for new foundations and new grounds for certainty.
一些澄清来自相对容易地揭示某些悖论的原因。特别是著名的理查德悖论，是一种认知行为的结果，这种行为在所有意图和目的上都是合法的：将可以用一种语言用有限数量的单词描述的所有数字聚集在一个类别中。但是这样的类是不存在的；它是不真实的，正如 Émile Borel 在 1908 年所说的那样，因为将“有限”属性分配给一组词以确保这些词想要描述的实体的实际存在是不够的。有限准确地说，必须是一个计算过程，从一组数据中能够在有限的阶段中详细说明结果。Borel 指出，数学实体的存在只能通过其实际构造来保证。例如，代数方程的实根可以使用能够评估小数总和达到所需准确度的程序来计算。因此，保证它存在的是算法。5并非偶然，Borel 被认为是系统研究算法的先驱，他将建立一门精确基于算法概念以及系统使用数值和信息程序的计算科学。6在 1908 年的同一年，Zermelo 发表了一项关于集合正确排序的研究，旨在确保它们可构造而没有悖论。不久之后，约翰·冯·诺依曼阐述了一个等价的理论。7
A few clarifications emerged from the relative ease with which it was possible to unmask the causes of certain paradoxes. The celebrated Richard’s paradox, in particular, was the consequence of a cognitive act that to all intents and purposes was legitimate: to aggregate in a single class all the numbers that may be described, in a language, with a finite number of words. But such a class does not exist; it is unreal, as Émile Borel would remark in 1908, because it is not enough to assign to a set of words the attribute ‘finite’ in order to ensure the actual existence of the entity that those words want to describe. Finite had to be, properly speaking, a process of calculation that from a set of data was capable of elaborating a result in a limited number of stages. Borel noted that the existence of a mathematical entity can only be guaranteed by its actual construction. The real roots of an algebraic equation, for instance, may be calculated using a procedure capable of evaluating the decimal sums up to the degree of accuracy required. That which guaranteed its existence, therefore, was an algorithm.5 Not by chance, Borel was considered the pioneer of the systematic study of algorithms that would have built a science of calculation based precisely on the concept of the algorithm and on the systematic use of numerical and information procedures.6 In the same year of 1908, Zermelo published a study of the proper ordering of sets that was meant to ensure that they were constructible free from paradoxes. Not long after this John von Neumann elaborated an equivalent theory.7
在 20 世纪的最初几年，通过数学直觉主义出现了对数值连续统理论的批判，其论点确实在当时，不直接依赖于寻找悖论解决方案的迫切需要。直觉主义的基本标准是只承认那些可以实际构建的实体是现存的和真实的。因此，有理数的算术不受直觉主义的批评，因为皮亚诺和戴德金的理论已经为自然整数的存在提供了充分的理由，这要归功于它们通过连续添加单位来有效构造。此外，戴德金用递归或归纳定义理论证明，数学的普通运算可以归因于一个连贯的和实际的存在，可计算性的概念是基于枚举的概念，即比如说，关于自然数序列。这诸如加法之类的运算的存在可以概括为这样一种想法：实际上可以通过一系列有限步骤将任意两个自然数相加——例如 3 + 5 = 8 或 346 + 512 = 858；一般来说，n + m代表每对自然整数n 和m。对此的基础是封闭的概念，根据它，人们可以看出对增长的一种控制。 数字和数量的集合，已经由古代数学提出。在从一个数到连续一个数的加法过程中，或者在两个自然数的加法或相乘中，无论多么大，一个数总是停留在同一域的范围内，即自然数的范围内，这对于这个原因被称为封闭。然而，我们需要指出一个关键的段落：即使闭包的性质有助于消除数字增长的危险，戴德金仍然没有注意到计算的不稳定性问题，这将成为最这种增长的阴险后果。很快，就不再足以向自己保证，尽管这些数字不成比例地增长，但仍然属于同一个领域。
In the first years of the twentieth century a critique of the theories of the numerical continuum emerged through mathematical intuitionism, with arguments that did not depend directly, at the time, on the urgent need to find a solution to paradoxes. The fundamental criterion of intuitionism was to acknowledge as extant and real only those entities that could actually be constructed. The arithmetic of rational numbers was consequently protected from intuitionist criticism, because the theories of Peano and Dedekind had given sufficient justification for the existence of natural whole numbers, guaranteed thanks to their effective construction by means of successive additions of units. Moreover, Dedekind, with the theory of recursion, or of definition by induction, had demonstrated that to the ordinary operations of mathematics a coherent and actual existence could be ascribed, and that the concept of calculability is based on that of enumeration, that is to say, on the series of natural numbers. The existence of an operation such as addition could be summarized in the idea that it is possible to actually add, with a series of finite steps, any two natural numbers – for instance 3 + 5 = 8 or 346 + 512 = 858; in general, n + m for every pair of natural whole numbers n and m. Fundamental to this was the idea of closure, in the light of which one could discern a kind of control to the growth of numbers and quantities, already posited by the mathematics of antiquity. In passing from one number to the successive one with the addition of a unit, or in the addition or multiplication of two natural numbers, no matter how large, one always remained in the ambit of the same domain, that of natural numbers, which for this reason was called closed. One needs, however, to signal a critical passage: even given that the property of closure served to remove the danger of the growth of numbers, Dedekind was still oblivious to the problem of the instability of calculation, which would turn out to be the most insidious consequence of that growth. Soon it would no longer be enough to assure oneself that, while growing disproportionately, the numbers still belonged to the same domain.
然而，实数的情况，至少在理论上，比自然数和有理数的情况更复杂，因为它们的定义意味着引入了无限类的数字以及系统地使用存在量词和全称量词，即说诸如“存在一个满足属性 P 的 x”或“对于每个x属性P 都是有效的”这样的命题。即使基于实数概念的经典分析似乎并未因发现自相矛盾而受到损害，但它仍然受到那些像直觉主义者一样不承认数学实体的人的批评和保留，数字，集合或函数实际上是不可计算的。
The case of real numbers, however, at least at a theoretical level, was more complex than that of natural and rational numbers, because their definition implied the introduction of infinite classes of numbers together with the systematic use of existential and universal quantifiers, which is to say of propositions such as ‘there is an x that satisfies the property P’ or ‘for every x the property P is valid’. And even if classical analysis, based on the concept of real numbers, did not seem to be compromised by the discovery of antinomies, it was still subject to criticisms and reservations on the part of those who, like the intuitionists, did not admit mathematical entities, numbers, sets or functions that were not actually calculable.
一旦用有理数建立了计算规则，实数就被定义为有理数的基本序列a 1，a 2，... ε无论多么小，都可以找到一个依赖于ε的自然数k ，对于每个自然数p ， a n + p和a n之间的距离（其中n > k）小于ε. 但这是什么意思，这个表达式可以找到 吗？直觉主义数学仅表明，如果n > k ，可以有效地计算a n + p和a n之间的距离小于ε的指数k，也就是说，存在一个能够计算该临界值的程序k的值。8这种简单的观察，布劳威尔和直觉主义者为了攻击经典分析而诉诸于这种简单的观察，足以使康托赖以提供一种包含现实与数学连续统的商。用符号表示基本序列不足以使对应于该符号的数字成为实数，除非可以计算出指数k ，通过该指数 k， a n + p和a n之间的距离在n > k时变得任意小。
Once the rules of calculation had been established with rational numbers, real numbers were defined as fundamental sequences a1, a2, … an, …, of rational numbers that satisfied, we emphasize again, the following property: if one assigns a number ε, however small, it is possible to find a natural number k, dependent on ε, for which the distance between an+p and an, where n > k, is less than ε for every natural number p. But what does it mean, this expression it is possible to find ? Intuitionist mathematics indicates only that it is possible to calculate effectively the index k for which the distance between an+p and an is less than ε if n > k, that is to say, that there exists a procedure capable of calculating that critical value of k.8 This simple observation, to which Brouwer and the intuitionists resorted in order to attack classical analysis, was sufficient to render fictitious precisely that atomistic concept of number which Cantor had relied on to lend a contained quotient of reality to the mathematical continuum. To denote a fundamental sequence with a symbol was not enough to make the number corresponding to that symbol real, unless it were possible to calculate the index k by which the distance between an+p and an becomes arbitrarily small for n > k.
对于直觉主义的数学家来说，有效微积分的理论基础与旨在以最有效的方式解决应用问题的算法科学所要求的具体性相去甚远。Brouwer 认为数学是一种行为，其现实性是基于意识和对随时间发展的多种感觉的直觉。实数的直觉主义构造，就像一个渐进的定义，随着时间的推移，包括它在内的越来越小的区间，甚至没有图灵、克林或冯诺依曼的微积分的重要性。计算数学，无论是理论的还是应用的，都将满足构建更加严格的效率标准的需求。
For the intuitionist mathematicians, effective calculus had theoretical bases that were far from the concreteness required by a science of algorithms orientated towards resolving problems of application in the most efficient way possible. Brouwer considered mathematics to be an action the reality of which was based on awareness and on the intuition of a multiplicity of sensations that developed over time. The intuitionist construction of a real number, conceived like a gradual definition, in time, of increasingly small intervals that include it, did not even have the materiality of the calculus of Turing, Kleene or von Neumann. Computational mathematics, theoretical and applied, would satisfy the demand for construction of ever more exacting criteria of efficiency.
需要通过程序有效地减小接近极限的近似值之间的距离，这不仅回应了如何定义无理数的理论问题。隐含的是对数学目标的新知识的需求。数学科学必须能够朝两个相反的方向前进：从具体到抽象，也从抽象到具体——至少不确定这两个方向中的哪个方向最能响应其最深刻和最隐秘的倾向. 实际上，至少有四个研究方向让我们重新思考几个世纪以来数学家们对他们的理论与现实世界（无论是物质的还是无形的）之间的关系所给予的极度关注：毕达哥拉斯学派和古代原子论者的数点、数学物理学、实数理论以及随后基于算法概念的微积分科学。
The need to render effectively small, by means of a procedure, the distances between approximations near to the limit did not only respond to a theoretical question of how an irrational number is to be defined. Implicit is the demand for a new knowledge of the objectives of mathematics. Mathematical science must be capable of going in two opposite directions: from the concrete to the abstract, but also from the abstract to the concrete – and it is at the very least uncertain which of the two directions responds best to its deepest and most secret inclination. In effect, at least four lines of research lead us to think again about the extreme attention that mathematicians have devoted over the centuries to the relation that their theories entertain with the real world, be it material or invisible: the doctrine of number-points of the Pythagoreans and of the ancient atomists, mathematical physics, the theory of real numbers and the subsequent science of calculus based on the concept of algorithm.
正如朱塞佩·皮亚诺（Giuseppe Peano）所观察到的，数学的目的主要在于计算数字的总和，这些数字是模拟自然界最多样化现象的方程的解：“数学的目的是确定表现出来的未知数的数值在实际问题中。牛顿、欧拉、拉格朗日、柯西、高斯和所有伟大的数学家都发展了他们奇妙的理论，直到计算所需的小数。 9现在，数学只有在面向具体和抽象的情况下才能完成这项任务。微分方程和积分方程，以及表示自然现象和人工现象的数学模型的最小值问题，起源于物理情况、经济和信息，以及边界值的补充条件的性质和未知函数的初始值总是受到人们试图模拟的物理现实的驱动。
As Giuseppe Peano observed, the aim of mathematics substantially consists of the calculation of sums of numbers that are solutions to the equations that simulate the most diverse phenomena of nature: ‘The aim of mathematics is to determine the numerical value of the unknowns that manifest themselves in practical problems. Newton, Euler, Lagrange, Cauchy, Gauss, and all the great mathematicians develop their marvelous theories up to the calculation of the required decimals.’ 9 Now, mathematics can carry out this task only if it is oriented towards the concrete as well as the abstract. The differential and integral equations, and the problems of the minimum, in which the mathematical models of both natural and artificial phenomena are expressed, have their origins in physical situations, economic and informational, and the nature of supplementary conditions on the boundary values and the initial values of the unknown functions is always motivated by the physical reality one is seeking to simulate.
然而，数学并不局限于寻找定义这些模型的方程。需要满足允许它详细说明解决方案的要求，并且该解决方案可以用解析公式表示的情况相对较少，就像假设存在解析公式也不太可能使用解析公式来找到根据全数字信息求解方程。
Mathematics does not limit itself, however, to finding equations that define these models. The requirements that allow it to elaborate a solution need to be satisfied, and it is relatively rare that this solution can be expressed with an analytical formula, just as it is also improbable that one should use an analytical formula, assuming it exists, to find the solution of an equation in terms of wholly digital information.
以分析形式编写的解决方案（如果存在）通常非常复杂。此外，解析公式必须满足稳定性要求：数据的微小变化，结果的微小变化必须对应；否则，模型传达的信息可能会被完全扭曲。这个条件并不总是得到满足，我们要向 Jacques Hadamard 提供一个初始值示例，该示例的解析解不以连续方式依赖于数据。只要这些受到最小错误的影响而没有与问题的解决方案有关的有用信息就足够了。这种类型的问题被称为“不适定”。10
The solutions written in analytical form, when they exist, are usually extremely complicated. Moreover, an analytical formula must satisfy the requirement of stability: to small variations of data, small variations in the results must correspond; otherwise, the information communicated by the model risks being completely distorted. This condition is not always satisfied, and we owe to Jacques Hadamard an example of initial values the analytic solution of which does not depend in a continuous way on the data. It is enough for these to be affected by a minimal error to have no useful information pertaining to the solution of the problem. Problems of this type have been termed ‘ill-posed’.10
方程的解存在并且是唯一的，并且问题应该被很好地提出的要求符合数学物理学的经典概念，该概念由物理事件以稳定和确定的方式演化的假设支配，一旦初步的周边条件已经建立。
The demand that the solution of an equation exists and is unique, and that the problem should be well posed, is in keeping with a classic conception of mathematical physics, dominated by the presupposition that a physical event evolves in a stable and determined way, once the initial surrounding conditions have been established.
拉普拉斯关于基于当前状态的详尽数据集计算物理世界整个未来的可能性的愿景是这种趋势的极端表现。尽管如此，这种以因果关系为基础的数学决定论的理性理想在面对物理现实时逐渐受到侵蚀。非线性现象、量子理论和强大的数值方法的出现表明，“适定”问题远非唯一能够反映真实现象的问题。11
Laplace’s vision of the possibility of calculating the entire future of the physical world on the basis of an exhaustive set of data on its current state is an extreme expression of this tendency. Nonetheless this rational ideal of mathematical determinism based on the idea of causality underwent a gradual erosion when confronted with physical reality. Non-linear phenomena, quantum theory and the advent of powerful numerical methods have shown that ‘well-posed’ questions are very far from being the only ones capable of reflecting real phenomena.11
但是仅仅保证分析解（假设它存在）对于数据是连续的是不够的。实际上，数学建模需要多个阶段：模型的公式通常需要通过算术公式来近似，而后者需要通过数值程序转换为纯数字计算。在最频繁在使用纯算术模型（例如线性代数方程组）来近似微分模型的情况下，需要检查人们对这个简化模型的数据误差的感知，以及过程的稳定性计算其数值解。最后，微积分包含大量基本运算和大量子程序，这些子程序构成了我们无法访问但计算过程的可信度在很大程度上取决于的隐藏细节。正如 Beresford Parlett 在 1978 年观察到的那样：
But it is not enough to assure oneself that the analytical solution, assuming it exists, is continuous with respect to the data. In reality, mathematical modelling entails various stages: the formula of the model needs to be approximated, usually, by an arithmetical formula, and the latter needs to be translated, by way of numerical procedures, into a purely digital calculation. In the most frequent case in which one approximates the differential model with a purely arithmetical one, such as a system of linear algebraic equations, one needs to examine one’s perception in relation to the errors of data of this simplified model, as well as the stability of the process that calculates its numerical solution. In the end, calculus encompasses an enormous accumulation of elementary operations and a multiplicity of subprograms that constitute a hidden elaboration to which we do not have access but on which to a large degree the credibility of a computational process depends. As Beresford Parlett observed in 1978:
这些天来，大多数指定的计算都是隐藏的。这意味着人类用户既看不到数据也看不到输出。在大型计算中，子任务（也许是傅立叶变换）的数据将由某个程序生成，而结果会被另一个程序迅速使用。这是内向数值分析的特征……隐藏计算的算法需要比人眼可以看到结果的算法更可靠。执行时间似乎不太重要，但需要可靠性和效率。在何种程度上，在每种情况下，我们可以两者兼得？这是一个有趣的问题。12
These days most well specified computations are hidden. This means that the human user sees neither the data nor the output. In a big calculation the data for a subtask (a Fourier transformation, perhaps) will be generated by some program and the results promptly used by another. This is characteristic of introverted numerical analysis … Algorithms for hidden computations need to be much more reliable than those for which results will be seen by a human eye. Execution time seems to be less crucial but both reliability and efficiency are wanted. To what extent, in each case, can we have both? That is an interesting question.12
概括地说，自莱布尼茨和牛顿数学时代以来，已经跨越了三个关键时刻，其中第三个目前仍在进行中，并呈现出我们习惯称为数字革命的轮廓。在 17 世纪第一次发现采用无穷小分析名称的宏大分析机器之后，在 19 世纪末开始了第二阶段，称为分析算术化，同样具有革命性：在数、集合和通向极限的概念中寻找分析的基础。在数量上，无论是理性的、真实的还是超限的，人们都需要寻找终极现实，即有限的真实现实，以及数学从未停止探索的无限。在这里也可以找到第三次革命的种子：从可以追溯与数概念相关的每一种推理形式的第一原理到实施各种计算集合的方法的运动数字。算法和计算出的数字列表将继承初始模型的信息，并在真正的还原论的支持下将其转化为不同的、更基本的、描述性的水平。
To summarize in broad terms, since the time of Leibniz and Newton mathematics has crossed three crucial moments, the third of which is currently still ongoing and is assuming the contours of what we are accustomed to refer to as the digital revolution. After the first discovery, in the seventeenth century, of that grandiose analytical machine that adopted the name of infinitesimal analysis, a second phase was initiated at the end of the nineteenth century, known as the arithmetization of analysis, equally revolutionary in kind: a search for the fundamentals of analysis in the concepts of number, of the set and of passage to the limit. In number, whether rational, real or transfinite, one needed to look for the ultimate reality, the authentic actuality of the finite, as well as of the infinite, for which mathematics had never ceased to search. Here, too, were to be found the seeds of the third revolution: the movement from first principles by which it was possible to trace back every form of reasoning relating to the concept of number to the implementation of a variety of methods for calculating sets of numbers. Algorithms and the lists of calculated numbers would inherit the information of initial models and would translate it, under the auspices of a genuine reductionism, on a different, more elementary, descriptive level. The theoretical possibility of returning analysis to the issue of number and to operations of the passage to the limit was to be developed alongside an actual arithmetization.
最后阶段是由于两个伴随的原因：数学基础的危机，这引发了人们对不分青红皂白地使用集合概念的可能性的怀疑，以及自 19 世纪末以来已经开始的应用科学的惊人发展，其影响之一是从 20 世纪中叶开始出现信息技术和科学计算。算法成为信息论的研究对象，而且从 20 世纪下半叶开始，大规模计算科学的理论和实践核心将数学物理问题转化为宏大的数字系统。阐述。
The final stage was due to two concomitant reasons: the crisis in the fundamentals of mathematics, which raised doubts about the possibility of indiscriminately using the concept of sets, and the surprising development of applied science, already under way since the end of the nineteenth century, which had among its effects the emergence, from the mid-twentieth century onwards, of information technology and scientific computation. Algorithms became an object of study for information theory, but also, from the second half of the twentieth century, the theoretical and practical nucleus of a science of computation on a large scale that would translate the problems of mathematical physics into a grandiose system of digital elaboration.
因此，在第三阶段，现实原则被转移到算法的概念上。它经常在这一转变过程中指出，数学不能将自己局限于抽象程序，还必须遵循相反的路径——从抽象概念到具体现实：可以说是到世俗的。从这个基础出发，从具体应用到物理现实，数学将获得力量，使自己摆脱危机的僵局，否则它会削弱其理论效力，就像安泰厄斯一样。巨大的安泰俄斯从与地球的接触中获得了力量，赫拉克勒斯成功地击败了他，将他举在空中，每次他跌倒时都将他抱起来。安泰俄斯就这样逐渐失去了力量，被打败并被杀死——但如果他与地球保持联系，他的力量肯定会占上风。
The principle of reality was consequently moved, in this third phase, on to the idea of algorithm. It was frequently noted during the course of this transition that mathematics cannot limit itself to procedures of abstraction but must also follow the opposite path – from abstract concepts to concrete reality: to the earthly, so to speak. From this very ground, from concrete applications to physical reality, mathematics was to derive the strength to free itself from the impasse of a crisis at its foundations that would otherwise have weakened its theoretical potency, Antaeus-like. The giant Antaeus derived his strength from his contact with the Earth, and Hercules managed to defeat him by holding him up in the air and picking him up every time he fell. Antaeus thus gradually lost his strength and was defeated and killed – but had he retained contact with the Earth, his strength would surely have prevailed.
Andrej A. Markov Jr 对算法思想的形式化做出了重要贡献，他特别求助于具体性原则：“抽象在数学中是必不可少的，但它们本身不应该作为目的而追求，也不应该导致他们不会下降到“地球”的点。我们应该始终认为从抽象思想到实践的转变是人类理解客观现实的必要时刻。13
Andrej A. Markov Jr, who made important contributions to the formalization of the idea of algorithm, had specific recourse to a principle of concreteness: ‘Abstractions are indispensable in mathematics, and yet they should not be pursued as ends in themselves, nor should they lead to a point at which they do not descend to “earth”. We should always think of the transition from abstract thought to practice as a necessary moment in human understanding of objective reality.’13
然而，随着 20 世纪的算法科学经历了一种新的抽象，虚拟实现的抽象，对抽象理论的研究使得在机器的物理空间和时间中发生的计算过程成为可能。马尔可夫指出，这种新的抽象首先在于使自己远离我们构建可能性的真正限制，并开始建立其理论前提。14实际可实现性首先被视为仅仅是虚拟的。实际上，在他的在关于递归函数的著名论文中，Hartley Rogers 宣称他问自己“计算机方法存在或不存在的问题，而不是效率或良好设计的问题”。15在 1937 年的一篇文章中著名地描述的图灵自己的机器中，人们面临着机械装置典型的具体性和符合可计算性的非物质概念的绝对数学抽象的奇特组合。通过具体程序的实际实现将取决于是否有足够的时间和空间资源的实际可用性。从 20 世纪下半叶开始，算法科学将致力于此，其任务是根据执行的空间和时间精确测量过程的计算成本。
Nevertheless, with the science of algorithms the twentieth century experienced a new kind of abstraction, the abstraction of virtual realization, the study of abstract theories that make possible the processes of computation that take place in the physical space and time of a machine. This new abstraction, noted Markov, consisted above all in distancing oneself from the real limits of our possibilities of construction, and in starting to establish its theoretical premises.14 Actual realizability was to be considered, in the first instance, as merely virtual. In effect, in his celebrated treatise on recursive functions, Hartley Rogers declared that he asked himself ‘questions of existence or non-existence of computer methods, rather than questions of efficiency or good design’.15 In Turing’s own machine, famously described in an article of 1937, one was faced with a peculiar combination of the concreteness that is typical of a mechanical contraption and the absolute mathematical abstraction that accords to an immaterial idea of calculability. A practical realization, by means of concrete procedures, would depend on the actual availability of sufficient resources of time and space. From the second half of the twentieth century a science of algorithms would be devoted to this, tasked with the role of measuring the computational cost of a procedure precisely in terms of space and time of execution.
15. 数字的增长
15. The Growth of Numbers
在希腊数学中，近似过程、数字的级数和几何图形的放大（或缩小）提出了关于它们无限延伸的严重问题。通过给ápeiron分配一个纯粹的潜在含义来解决这些困难，就像在穷举法或交互式算法中一样. 在这些算法中，计算的数字随着数字的增长或减少而增长，并且由于准确性需要越来越大的数字，因此从某个点开始，计算将变得完全不可能。
In Greek mathematics the processes of approximation, the progression of numbers and the enlargement (or reduction) of geometrical figures posed serious questions regarding their prolongation to infinity. The difficulties were resolved by assigning to the ápeiron a purely potential meaning, just as in the methods of exhaustion or in interactive algorithms based on the repeated, indefinite application of an operator to the approximate value of the solution of a problem, updated step by step. In these algorithms the calculated numbers grew in correspondence with growth or decrease of the figures, and since accuracy would have required increasingly larger numbers the calculation would have become, from a certain point onwards, altogether impossible.
此外，在希腊数学中，不存在大型自动计算，审议处理的数字相对较小。只有阿基米德，在《算沙者》中，想到了一种方法来计算更大的数字，数以百万计的总和，足以计算出足以填满整个宇宙的沙粒。但是这些数字虽然非常大，但几乎不能与无穷大相比，而且最重要的是，它们并不构成真正计算的一部分。因此，在阿基米德的论文中，人们无法收集到任何关于它们的成长的全神贯注，而且它们的实际存在也没有受到质疑。
Moreover, in Greek mathematics large automatic calculation did not exist, and deliberations dealt with relatively small numbers. Only Archimedes, in The Sand Reckoner, thought about a method of conceiving of much larger numbers, of millions of millions of sums, sufficient to count enough grains of sand to fill the entire universe. But these numbers, albeit extremely large, were scarcely comparable to the infinite, and above all did not form part of real computation. Consequently, in Archimedes’ treatise one cannot glean any preoccupation with their growth, and their actual existence was not questioned.
在现代计算科学中，操作数量有限的程序也会出现困难，由于对有限性本身无法保证的现实程度的需求。一般而言，使用p / q形式的有理数的自动计算（其中p和q是整数）在其运算的最基本阶段具有问题特征：不能排除几次乘法将产生大量数字的可能性超出计算器内存空间的限制。
In the modern science of computation, difficulties also arise in procedures with a limited number of operations, due to the demand for a degree of actuality that the finite, in and of itself, is not capable of guaranteeing. In general, automatic computation with rational numbers of the form p/q, where p and q are whole numbers, has a problematic character from the most elementary phases of its operations: one cannot rule out the possibility that a few multiplications will produce enormous numbers that exceed the limits of space in the calculator’s memory.
另一个问题出现了：用分数来逼近一个无理数实际上有多接近？几何形状的放大或缩小与数字的增长相对应。在使用牛顿迭代法逼近整数的平方根的情况下提供了一个示例性实例，由此近似分数的分子和分母之和的数量每次与构造的平方序列相关时加倍与连续的晷针校正。横向和对角线数也在增长，尽管速度较慢。
A further question arises: how closely is it actually possible to approximate an irrational number by way of a fraction? The enlargement or diminution of geometrical shapes has as its counterpart the growth of numbers. An exemplary instance is provided in the case of the approximation of the square root of a whole number with Newton’s iterative method, whereby the number of sums of the numerator and denominator of the approximate fractions is doubled every time in correlation with a sequence of squares constructed with successive gnomonic corrections. The lateral and diagonal numbers also grow, albeit more slowly.
但是将迭代延长多长时间才有意义？原则上，分数的分子和分母可以无限增长，以尽可能提高无理数的逼近。然而，无理数和近似分数之间的距离逐渐减小是有客观限制的。Joseph Liouville 在 1844 年证明了一个定理，建立了代数无理数逼近的理论极限，也就是说，一个数是代数方程的解，通过将具有整数系数的多项式等于 0 获得：逼近的误差，即之间的距离随着q的增长，代数数和近似分数p/q并没有像我们希望或期望的那样减少。但刘维尔定理有大量不同类型的含义——从超越数的构造，即非代数数，到在逼近代数数的过程中测量数字的增长速度。在任何情况下，数字的增长都是为逐步解决问题所付出的代价。
But for how long does it make sense to prolong the iteration? In principle, the numerators and the denominators of the fractions could grow indefinitely, in order to improve as far as possible the approximation of the irrational number. Nevertheless, there are objective limits to the progressive diminution of the distance between the irrational number and the approximating fraction. With a theorem demonstrated in 1844, Joseph Liouville established the theoretical limit for the approximation of an algebraic irrational number, that is to say, of a number that is the solution of an algebraic equation, obtained by equating to 0 a polynomial with integer coefficients: the error of approximation, that is to say, the distance between the algebraic number and an approximating fraction p/q does not diminish as much as we might desire or expect as q grows. But Liouville’s theorem has a wealth of implications of a different kind – from the construction of transcendental numbers, that is to say, non-algebraic ones, to the measurement of the speed of growth of numbers during the process of approximating an algebraic number. In every case the growth of numbers is the price paid for the gradual closing in on the solution to the problem.
更准确地说，一个数是k次代数的，如果它是一个代数方程的解，它是通过将 k 次多项式而不是低于 k 次的多项式等于0获得的。例如，
More precisely, a number is algebraic to a degree k if it is the solution to an algebraic equation obtained by equating to 0 a polynomial of degree k, but not a polynomial of a lower degree than k. For example, 2 is an algebraic number of degree 2 because it is the solution of the degree 2 equation x 2 − 2 = 0, and is not the solution to any linear equation (that is to say, of degree 1). Liouville’s theorem states that for every algebraic number x of degree n > 1 the growth of q does not result in a lessening, beyond a certain limit, of the distance from x of an approximate fraction p/q. This distance remains larger than 1/q n + 1 for denominators q that are sufficiently large. For example, the difference in absolute value between an irrational number defined by the square root of a positive whole number and its approximate fraction p/q remains higher, for a sufficiently large q, to 1/q 3. The numbers that are not solutions to any algebraic equation, transcendental numbers such as e and π, are not subject to this limitation. There is a non-numerable infinity of transcendental numbers, because as Cantor demonstrated the algebraic numbers form a numerable set, and the real numbers, including algebraic numbers and transcendental numbers, form a set of greater power than the numerable. Thanks to his theorem, Liouville was able to demonstrate for the first time how it is possible to construct transcendental numbers. Liouville’s transcendental numbers take the form:
a = 0. a 1 a 2 000 a 3 00000000000000000 a 4 000000000. . . . . .
a = 0.a1a2000a300000000000000000a4000000000. . . . . .
其中a i是 1 到 9 之间的十进制数，0 组的长度增长非常快（具有阶乘速度，因此大于指数速度）。现在，假设a是n次的代数数，如果考虑用足够数量（取决于n ）的 0 组获得的a的近似值b，我们得到a和b之间的距离，该距离与刘维尔定理不相容. 因此a不可能是代数数。在这种情况下，正是 0 序列长度的非常快速的增长，使得它能够确定：a是超越的。1
where ai is a decimal number between 1 and 9, and the groups of 0s have a length that grows very rapidly (with a factorial speed, therefore greater than an exponential one). Now, supposing that a is an algebraic number of degree n, if one considers an approximation b of a obtained with a sufficient number (depending on n ) of groups of 0s, we obtain a distance between a and b that is incompatible with Liouville’s theorem. Therefore a cannot be an algebraic number. In this case, it is precisely the very rapid growth of the length of sequences of 0s that enables it to be established that a is transcendental.1
刘维尔定理还可以测量用迭代算法逼近的分数的分母的增长速度，代数数z ：该方法的收敛速度越快，增长越快，即也就是说，计算的分数越快越接近z。同样的定理还可以找到分母的大小与计算它所需的操作数之间的关系。2
Liouville’s theorem also makes it possible to measure the speed of growth of the denominator of the fractions with which it approximates with an iterative algorithm, an algebraic number z : the growth is all the more rapid the greater the rapidity of convergence of the method, that is to say, however more rapidly the calculated fractions get nearer to z. And the same theorem also makes it possible to find a relation between the size of the denominator and the number of operations required to calculate it.2
也就是说，数字增长的主要不便是另一回事。在数字计算中，数字由有限的二进制数序列（0 和 1）组成。但是，每次使用这些序列（其位数不能超过固定限制）进行操作时，都会产生舍入错误，这可能会导致数值不稳定并导致确定所需信息的致命损失以足够的精度解决问题。扰动理论是数字微积分的重要阶段，对于动态系统和确定性混沌理论也是如此。由一个离散的动态系统模拟的现象行为的可预测性条件，由一个简单的迭代定律定义，在于它对误差的敏感性。如果我们承认相对于它开始演化的初始条件有多么小的扰动，那么关于一个现象将如何发展的预测可能完全不可靠。
That said, the principal inconvenience of the growth of numbers is a different one. In digital calculation, numbers consist of finite sequences of binary numbers (0 and 1). But with every operation using these sequences, whose number of digits cannot exceed a fixed limit, an error of rounding is produced that can generate numerical instability and a fatal loss of the information necessary to ascertain the solution to a problem with a sufficient degree of precision. The theory of disturbance is an essential stage in digital calculus, as well as with regard to the theory of dynamic systems and of deterministic chaos. A condition for the predictability of the behaviour of a phenomenon simulated by a dynamic system which is discrete, defined by a simple iterative law, resides in its sensitivity to error. Prediction as to how a phenomenon will develop may be altogether unreliable if we admit perturbations, however small, with respect to the initial conditions from which it begins to evolve.
使用p / q形式的有理数进行计算通常是不切实际的。分子p和分母q通常增长过快，以至于超出了机器内存的限制。数字的增长还体现在计算复杂度上：这些数字越大，计算p和q的成本就越高。第一个补救措施是放弃以p / q形式表示有理数，并使用 浮点数(. a 1 a 2 ... a n ) B k，其中n是预先确定的数字，有关数字大小的信息位于指数k上。仅在少数特殊情况下，采用传统的分数计算是有利的，与其说是通过将p和q划分为最终的公因数而获得的简化，不如说是连续近似和特殊舍入技术的结果。3然而，这一认识并不足以消除因人数或业务异常增长而造成的困难。因此，增长可能会导致所有声称的现实从人们试图计算的东西中消失。
Computation using rational numbers of the form p/q is generally impractical. The numerator p and the denominator q usually grow too rapidly, to the extent that they exceed the limits of the machine’s memory. The growth of numbers is also reflected in computational complexity: the calculation of p and q is all the more expensive the larger these numbers become. The first remedy consists in abandoning the representation of a rational number in the form p/q, and in the use of floating point numbers (.a1 a2 … an )B k, where n is a predetermined number and information on the size of the number is located on the exponent k. Only in a few particular cases will it be advantageous to resort to traditional calculation with fractions, not so much by way of simplifications obtained by dividing p and q for eventual common factors, but rather as a result of continuous approximations and special techniques of rounding.3 This realization is not enough, however, to dispel the difficulties due to the abnormal growth of numbers or operations. Growth may consequently cause the disappearance of all purported reality from what one is trying to calculate.
正如唐纳德·克努斯 (Donald Knuth) 所写，“分数计算的经验表明，在许多情况下，数字会变得非常大......在每个加法、减法、乘法和除法子例程中溢出。4
As Donald Knuth writes, ‘experience with fractional calculations shows that in many cases the numbers grow to be quite large … it is important to include tests for overflow in each of the addition, subtraction, multiplication, and division subroutines’.4
数学通常不专注于计算数字，无论它们有多大或有多大。如果这样做，那只是出于实际或应用的需要。尽管如此，如何做到这一点，如何做到这一点的问题一个人可能会计算出描述世界所不可或缺的实际数字，这会提出具有显着理论意义的问题。将微分和积分模型大规模转化为算法和数字计算，最终成为 19 世纪末分析算术化的当代版本，导致对基础研究面临的重大问题进行重新表述在 20 世纪上半叶：一个问题或一类问题何时以及以何种方式承认解决方案？真正解决问题意味着什么？自动程序中的有限和无限有什么意义？连续和离散之间有什么关系？
Mathematics does not occupy itself, usually, with calculating numbers, regardless of how small or big they might be. If it does so, it is only out of practical or applied necessity. Nonetheless, the question of how to do it, of how one might calculate the actual numbers that are indispensable for a description of the world, poses problems of notable theoretical interest. The translation of differential and integral models into algorithms and into digital computation on a large scale, which is ultimately the contemporary version of the arithmetization of analysis at the end of the nineteenth century, leads to a reformulation of the great issues faced by research into fundamentals in the first part of the twentieth century: when and in what way does a problem or a class of problems admit a solution? What does it mean to actually resolve a problem? What meaning does the finite and the infinite have in automatic procedures? What relation is there between the continuous and the discrete?
在 40 年代初期，数值计算，包括用数值而不是纯粹的解析术语求解方程，5仍处于相对初级的状态：它更像是一门艺术而不是计算科学。在某种程度上，这种情况是自相矛盾的，因为自古巴比伦微积分以来，也就是说自公元前1800 年以来，数学一直基于纯算术和数值类型的算法，正如我们在严格几何理论中发现的部分预兆欧几里得元素。
In the early forties, numerical computation, which consists of solving an equation in numerical rather than purely analytical terms,5 was still in a relatively rudimentary state: it was more of an art than a science of computation. To a certain extent the situation was paradoxical, because mathematics since the Babylonian calculus of antiquity, that is to say since 1800 BC, had been based on algorithms of a purely arithmetical and numerical kind, as we find prefigured in part in the rigorously geometric theory of Euclid’s Elements.
用于逼近无理数的程序，例如基于antanaíresis的程序，具有迭代形式：近似值是用递归公式计算的，这些公式在数字上再现了几何形状的增长。在代数上，欧几里得算法实际上与分数的构造有关，即所谓的连续分数，能够逼近无理数，而实数可以用连续分数来定义。但是分数的分子和分母的递归生成通常会产生不受控制的增长现象，从而导致计算停止。
The procedures that were used to approximate irrational numbers, such as those based on antanaíresis, had an iterative form: the approximate values were calculated with recurrent formulas that reproduced numerically the growth of geometric shapes. Algebraically, the Euclidean algorithm was virtually linked to the construction of fractions, the so-called continuous fractions, capable of approximating irrational numbers, and the real numbers may be defined in terms of continuous fractions. But the recursive generation of the numerators and denominators of fractions often produces phenomena of uncontrolled growth that can bring a halt to computation.
令人惊讶的是，最先进的计算继承了古代数学的模式，而后者又受到宗教的启发。但是，通知它的标准也对程序的效率进行了分析。根据数字计算的最新需求，古代公式，当属于递归类型时，已经是机械程序。但是直到现在我们才意识到递归过程的每个循环，无论多么简单，都可以继承之前的错误。6因此，即使是构造分数以逼近无理数的最基本的循环关系也暴露于数值不稳定的现象，从而导致不真实，这是致命的。康托尔和戴德金认为实际存在的相同数字实体。
The most advanced kind of computation has inherited, surprisingly, the schema of ancient mathematics, which was inspired in turn by religion; but the criteria that inform it have also imposed an analysis of the efficiency of the procedures. Ancient formulas, when of a recursive kind, were already mechanical procedures, in accordance with the most recent demands of digital calculation. But only now are we conscious of the fact that every cycle of a recursive process, however simple, can inherit the errors made in preceding ones.6 And it is therefore fatal that even the most elementary recurring relations with which fractions are constructed to approximate irrational numbers are exposed to phenomena of numerical instability that render unreal the very same numerical entities that Cantor and Dedekind thought actually existed.
也简化为循环关系的是特殊函数的演算，例如所谓的贝塞尔函数，以及特定类别的数字，例如伯努利数。Jakob Bernoulli 在他的Ars Conjectandi (1713)中首次介绍了后者，Leonhard Euler 随后更深入地研究了它们。这些数字涉及近似积分的简单离散和，因此涉及关于连续与离散之间、算术计算与连续之间关系的问题自芝诺悖论以来，在数字和几何量级之间的现象一直是数学研究的标志。它们的性质，以及它们与数字π的奇异关系，证明了对具有相当大的理论兴趣的单独研究是正确的。伯努利数是有理数，分数的分母是明确知道的，但分子不知道，它会迅速增加。为了计算它们，最近设计了复杂的算法：B 200的分子的五个质因数中的两个，分别为 90 位和 115 位的 200 位伯努利数，已经通过最强大的数字计算器进行了评估。欧拉一定已经对伯努利数增长的速度感到惊讶。他评论说，这些形式是“一个快速发散的序列，比任何其他递增项的几何序列都具有更大的力量”。7
Also reduced to recurrent relations is the calculus of special functions, such as the so-called Bessel functions, as well as particular classes of numbers such as Bernoulli numbers. Jakob Bernoulli first introduced the latter in his Ars Conjectandi (1713), and Leonhard Euler subsequently studied them in greater depth. These numbers are implicated in simple discrete sums that approximate integrals and consequently in questions regarding the relation between the continuous and discrete, between arithmetical calculations and continuous phenomena, between numbers and geometric magnitudes, that have marked mathematical research ever since Zeno’s paradoxes. Their properties, as well as their singular relation with the number π, justify a separate study of considerable theoretical interest. Bernoulli numbers are rational numbers, fractions of which the denominators are explicitly known but not the numerators, which increase rapidly. In order to calculate them, in recent periods complex algorithms have been devised: two of the five prime factors of the numerator of B200, the two-hundredth Bernoulli number, of 90 and 115 digits respectively, have been evaluated thanks to the most powerful digital calculators. Euler must already have been astonished by the speed at which Bernoulli numbers grow. These form, he remarked, ‘a sequence that diverges rapidly and grows with greater force than any other geometrical sequence of increasing terms’.7
16. 矩阵的增长
16. The Growth of Matrices
应用数学和计算数学中的每个问题最终都可以追溯到解决线性方程组的目标。这是一个论文，或者更确切地说是一个观察，现在已被广泛接受。如果问题不是线性的，它们就会变成线性的，通过这种方式，我们以近似的方式推导出关于求解微分方程或积分方程、最小或多项式逼近问题的第一个有用信息。近似不是缺陷，它是规范，一切都取决于量化它，以确定何时可能和方便。线性化意味着近似中的错误，但并不比从初始方程到纯数字计算的各个阶段所需要的其他错误更致命。
Every problem in applied and computational mathematics can be traced back in the end to the aim of resolving a system of linear equations. This is a thesis, or rather an observation, that is by now widely accepted. If the problems are not linear, they become linearized, and in this way we derive, in an approximate way, the first useful information on the solution of a differential or integral equation, of a problem of the minimum or of polynomial approximation. The approximation is not a flaw, it is the norm, and everything depends on quantifying it, in establishing when it is possible and convenient. The linearization implies an error in the approximation, but one not more fatal than others that are entailed in the various stages that lead from the initial equations to pure digital calculation.
从 20 世纪下半叶开始，很明显，对应用有用的线性方程组具有非常大的维度，并且假设线性方程组由系数矩阵标识，也就是说，由一张按行列排列的数字表，在见证数字计算最早发展的几十年中，迫切需要对大规模矩阵计算进行深入研究。现在这是主要困难：重新连接初始模型，即微分或积分方程，与求解线性方程组所需的纯数学计算。“线性”这个词不应该欺骗我们。线性和非线性的区别有在计算和应用数学中，通常标记为可解决性的关键边界，但存在绝对抵制任何解决尝试的线性问题。当线性方程组的系数矩阵具有很强的病态条件时，通常会出现这种情况，也就是说，当系统的解与数据中的误差有关时，有很高的意识。
From the second half of the twentieth century onwards it became clear that the systems of linear equations that were useful for applications have very large dimensions, and given that a system of linear equations is identified by a matrix of coefficients, that is to say, by a table of numbers arranged in rows and columns, in the decades that witnessed the earliest development of digital computation an in-depth study of matrix calculation on a large scale was urgently required. Now this was the principal difficulty: to reconnect the initial model, that is to say, the differential or integral equation, with the purely mathematical calculations necessary to solve a system of linear equations. The term ‘linear’ should not deceive us. The distinction between linear and non-linear has often marked, in computational and applied mathematics, a critical border of resolvability, but there are linear problems that are absolutely resistant to any attempt at resolution. This is typically the case when the matrix of coefficients of a system of linear equations is strongly ill conditioned, that is to say, when there is a high awareness of the solution of the system in relation to errors in the data.
n行m列的矩阵 A是一个排列成矩形的数字表，其中位置 ( i , j ) 的元素由第i行和第j列组成，用符号a ij 表示。如果n = m，矩阵为正方形。矩阵是自给自足的数学实体，就像数字、多项式和函数一样，矩阵的代数基于求和、乘积和求逆运算，类似于数字之间的运算，但通常没有某些属性，例如乘积的交换性，适用于数字。矩阵的元素可以是线性方程组的系数，而方程的已知项排列在定义向量的数字列中。
A matrix A of n rows and m columns is a table of numbers arranged in a rectangle, where the element of position (i, j ), consisting of the row i and column j, is denoted by the symbol aij . If n = m, the matrix has a square form. Matrices are self-sufficient mathematical entities, just like numbers, polynomials and functions, and the algebra of matrices is based on sum, product and inversion operations, analogous to those between numbers but usually devoid of certain properties, such as commutativity of the product, that apply to numbers. The elements of a matrix may be the coefficients of a system of linear equations, while the known terms of the equations are arranged in a column of numbers that defines a vector.
在当前的符号化中，矢量通常表示为箭头，具有自己的方向和长度，但也可以将矢量想象成一列数字。在笛卡尔平面上，一个向量v可以用一个只有两个数字x和y的列来表示，对应于它在轴上的投影。
In current symbolization a vector is usually represented as an arrow, with its own direction and length, but it is also possible to conceive of a vector as like a column of numbers. On the Cartesian plane a vector v with one may be represented with a column of only two numbers x and y, corresponding to its projections on the axes.
因此，在具有n维的空间中，向量v被简单地定义为包含n 个数字的列。维数不一定以空间或时空概念为前提。引入更多维度可能只是用数学术语思考问题的一种方式。在经济学中，可能有必要将支出划分为工业部门。假设有十个部门（汽车、纺织、海军、建筑等），支出将由一个由十个组成部分组成的向量表示，每个组成部分都是衡量特定部门成本的数字。
In a space with n dimensions a vector v is consequently defined as simply a column of n numbers. The number of dimensions does not necessarily presuppose an idea of space or space-time. To introduce more dimensions may simply be a way of thinking about the problem in mathematical terms. In economics it may be necessary to configure an expenditure divided into industrial sectors. Given, let’s say, ten sectors (automobile, textile, naval, construction, etc.), the expenditure will be represented by a vector of ten components, each one of which is the number that measures the cost in a particular sector.
图 8
Figure 8
方阵A对向量v的作用类似于将v与另一个向量w相关联的运算符或变换。然后我们写w = Av，其中w等于A和v的乘积。1这种情况变得很重要，在矩阵算子A的作用下，向量v不改变方向，也就是说，它被转换为向量w ，该向量具有v的相同分量乘以相同的实数或复数λ. 因此我们把它写成Av = λ v，一个表达一种不变性的公式，因为在算子A的作用下向量的方向保持不变。在这种情况下，λ是矩阵A的特征值，而v是相应的特征向量。在许多情况下，矩阵的特征值决定了求解线性方程组Ax = b（其中b是已知项的向量）的方法的效率程度，以及对解的认识X关于数据A和b的错误。
A square matrix A acts on a vector v like an operator or a transformation that associates v with another vector w. We then write w = Av, with w being equal to the product of A and v.1 The case then becomes important in which, under the effect of a matrix operator A, the vector v does not change direction, that is to say, it is transformed into a vector w that has the same components of v multiplied by the same real or complex number λ. We thus write this as Av = λv, a formula that expresses a kind of invariability, because the direction of the vector remains unchanged under the effect of the operator A. In which case λ is an eigenvalue of the matrix A, while v is the corresponding eigenvector. It is the eigenvalues of a matrix, in many cases, that decide the degree of efficiency of a method for resolving a system of linear equations Ax = b (where b is the vector of the known terms), as well as awareness of the solution x with respect to the errors for the data A and b.
向量和矩阵的代数最初是在几何和代数的背景下发展起来的，但随着时间的推移矩阵计算的影响在应用数学的不同领域变得越来越明显，在平衡、力学、理论物理、电子工程和计算的研究中，例如在与在线搜索引擎的数值计算方面的交互中。
The algebra of vectors and matrices was initially developed in the context of geometry and algebra, but over time the implications of matrix calculation became ever more evident in different sectors of applied mathematics, in the study of equilibria, mechanics, theoretical physics, electronic engineering and computing, for example in interactions with the numerical-computational aspects of online search engines.
计算的真正新颖性，从四五十年代开始，源于问题的更高维度，以及旨在以足够精度模拟物理现象的线性方程组的系数矩阵具有数万行的事实和列。因此，这种系统的解决需要一个自动程序，该程序考虑到数字表示技术、执行时间和所需的内存空间。这就要求对用于计算的算术定律进行彻底的改变。
The true novelty of computation, starting from the forties and fifties, derived from the heightened dimension of problems and from the fact that the coefficient matrices of a system of linear equations that aims to simulate a physical phenomenon with sufficient precision has tens of thousands of rows and columns. The resolution of such a system hence requires an automatic procedure that takes into account the techniques of the representation of numbers, the time taken to execute and the space of memory that is required. This demands a radical change in the arithmetical laws with which calculations are made.
矩阵代数反过来成为数字计算的核心章节。几乎所有的数学计算问题都可以追溯到线性代数和矩阵计算。2 Gilbert Strang 在这个话题上的话非常有说服力：“对于工程师以及社会和物理科学家来说，线性代数现在占据了一个通常比微积分更重要的位置。我这一代的学生，当然还有我的老师，都没有看到这种变化的到来。部分原因是从模拟到数字的转变，其中功能被矢量取代。线性代数将n维空间的洞察力与矩阵的应用相结合。3
The algebra of matrices became in turn a central chapter in digital calculation. To linear algebra and matrix calculation almost all the computational problems of mathematics can be traced.2 Gilbert Strang’s words on this topic are extremely telling: ‘For engineers and social and physical scientists, linear algebra now fills a place that is often more important than calculus. My generation of students, and certainly my teachers, did not see this change coming. It is partly the move from analog to digital, in which functions are replaced by vectors. Linear algebra combines the insight of n-dimensional space with the applications of matrices.’3
在 40 年代大规模微积分发展的第一阶段，经常查阅并经历了几个版本的标准手册是 Whittaker 和 Robinson 的The Calculus of Observations。但课文主要关注的是插值问题、非线性方程的解析、数值平方和傅里叶分析。线性方程组的分辨率并没有在本身分析的问题中体现出来。它只是基于最小二乘技术的近似理论的功能附录，唯一被推荐并在几页中描述的技术是克莱默设计的技术。这种方法完全无效的性质很快就被充分阐明了。
In the first phases of the development of calculus on a large scale, during the forties, a standard manual that was frequently consulted and went through several editions was Whittaker and Robinson’s The Calculus of Observations. But the text was largely concerned with problems of interpolation, on the resolution of non-linear equations, on numerical squaring and on Fourier analysis. The resolution of a system of linear equations did not figure among the problems analysed per se. It was only a functional appendix to a theory of approximation based on the technique of least squares, and the only technique that was recommended, and described in a few pages, was the one devised by Cramer. The utterly ineffectual nature of this method was soon enough made abundantly clear.
20 世纪微积分科学的先驱之一詹姆斯·威尔金森（James Wilkinson）也查阅了惠特克和罗宾逊的手册，他在 1946 年至 1948 年间与艾伦·图灵一起在国家物理实验室工作。在此期间，在军备研究部， Wilkinson 的任务是求解具有 12 个未知数的 12 个线性方程——正如他自己描述的那样，他最初的尝试使用了 Cramer 方法。这可能是第一次使用有限数量的操作的递归方法表明自己绝对无法实现。造成这种情况的原因不是问题而是方法，它需要算术运算的阶乘数，因此需要计算时间，仅对 50 个方程 - 大约 10 -6执行乘法的秒数——大于宇宙从假设的大爆炸到现在的时间。很明显，一个需要如此天文数字尽管有限的时间来计算的解决方案根本不是解决方案，而且在某种意义上它是完全不真实的。其原因是方程数量n的增加，算术运算数量的异常而非指数增长。4
Whittaker and Robinson’s manual was also consulted by James Wilkinson, one of the pioneers of the science of calculus in the twentieth century, who worked alongside Alan Turing at the National Physical Laboratory between 1946 and 1948. During this period, in the Armament Research Department, Wilkinson was given the task of solving twelve linear equations with twelve unknowns – and his initial attempt, as he describes it himself, made use of the Cramer method. This was perhaps the first time in which a recursive method, with a finite number of operations, revealed itself to be absolutely unattainable. The cause of this was due not to the problem but to the method, which requires a factorial number of arithmetical operations and hence a calculation time, for only fifty equations – with about 10−6 seconds to execute a multiplication – that is greater than the time between the birth of the universe from a hypothetical Big Bang to the present day. It was evident that a solution that would take such astronomical albeit finite time to calculate was no solution at all, and that it was in a certain sense completely unreal. The reason for this was the increase in the number n of equations, the abnormal rather than exponential growth of the quantity of arithmetical operations.4
1894 年，David Hilbert 发表了一篇关于使用最小二乘准则通过多项式逼近连续函数的文章。5面临的问题希尔伯特对应于一种基本的计算策略：将任何（连续）函数的评估带回到一个简化的微积分模型，例如仅由加法和乘法组成的多项式模型。自从牛顿代数以来，数学一直在寻求这种简化，该代数旨在用 x 的幂的简单组合来表示具有显着分析复杂性的函数f ( x ). 在 19 世纪，在希尔伯特之前，许多数学家都致力于这个主题，并且在 1885 年，魏尔斯特拉斯证明了定义在闭区间中的每个连续函数都可以用多项式统一逼近。因此，除了在任何情况下都不可避免的逼近误差外，至少在原则上，连续函数的演算被简化为一个有限数，这取决于逼近多项式的等级，仅是求和和乘法。这也可以容纳在更广泛的算术分析项目中。
In 1894 David Hilbert published an article on approximation, using the criterion of least squares, of a continuous function by a polynomial.5 The problem confronted by Hilbert corresponded to a fundamental computational strategy: to bring back the evaluation of any (continuous) function to a simplified model of calculus, such as that of a polynomial, consisting exclusively of additions and multiplications. Mathematics has always sought such simplifications, ever since Newton’s algebra, which aimed to represent functions f (x ) of notable analytical complexity with simple combinations of powers of x. In the nineteenth century various mathematicians had devoted themselves to this topic before Hilbert, and in 1885 Weierstrass had demonstrated that every continuous function defined in a closed interval allows itself to be uniformly approximated by a polynomial. Therefore, except for an error of approximation that is in any case inevitable, the calculus of the continuous function was reduced, at least in principle, to a finite number, dependent on the grade of the approximating polynomial, of sums and multiplications only. This could also be accommodated in the more extensive project of arithmetizing analysis.
现在希尔伯特的问题需要求解一个线性方程组，其中系数矩阵H为n行n列的平方：
Now Hilbert’s problem required the resolution of a system of linear equations in which the matrix H of the coefficients, square with n rows and n columns:
这在今天被称为数值难以处理的矩阵模型。矩阵H现在称为希尔伯特矩阵。
which is known today as the model of a numerically intractable matrix. The matrix H is now called the Hilbert matrix.
上个世纪计算机科学之父之一乔治·福赛斯不得不与希尔伯特打交道几十年后的矩阵。在 1970 年的一篇重要文章中，他观察到，对于维数n几乎不大于 8 或 9，任何自动程序都无法求解具有 n 个由 H 定义的未知数的n线性方程组，并且对于较小的值n，例如n = 6，在略有不同的机器上计算的明显相同问题的解决方案之间存在很大差异。我们现在认为，这些困难的原因在于矩阵非常病态。. 现在，如果输入数据的微小变化会在解决方案中产生很大的变化，则将问题描述为病态问题，而与计算中使用的程序无关。
One of the fathers of computer science in the last century, George Forsythe, had to deal with the Hilbert matrix several decades later. In an important article of 1970 he observed that, for a dimension n barely greater than 8 or 9, any automatic procedure does not have the capacity to solve a system of n linear equations with n unknowns defined by H, and that for small values of n, such as n = 6, there was a large discrepancy between the solutions of apparently identical problems calculated on machines that were just slightly different. We would now argue that the reason for these difficulties resides in the fact that the matrix is very ill conditioned. Now, a problem is described as ill conditioned if small variations in the input data can produce great variations in the solution, regardless of the procedure used in the calculation.
希尔伯特研究的近似过程包括使由于用多项式替换连续函数而导致的误差表达式最小化。为了使误差最小化，我们必须求解一个线性方程组Hx = b，其矩阵H即使对于小维度也是难以处理的。6对于n = 6， H的逆H -1的元素已经
The procedure of approximation studied by Hilbert consists of making minimal an expression of the error due to the substitution of a continuous function by a polynomial. To minimize the error we must then solve a system of linear equations Hx = b, the matrix H of which is intractable even for small dimensions.6 Already for n = 6 the elements of the inverse H −1 of H
引导我们预见它们的不可估量的增长，即使n的值稍微大一点。7现在，数字4410000 在H的倒数的位置 (5, 5)具有以灾难性方式放大已知项向量b的第五个元素上的扰动的能力，即使它非常小。8
lead us to foresee their immeasurable growth, even for values of n that are just slightly greater.7 Now, the number 4410000 in the position (5, 5) of the inverse of H has the capacity to amplify a perturbation in a catastrophic way, even if it is very small, on the fifth element of vector b of the known terms.8
对于n = 6，数字 4410000 是矩阵H - 1的最大模元素；但是如果我们允许n增长，关于x元素的误差会变得越来越大。对于n = 10，逆的最大元素（以模数表示）为 3.48 × 10 12，因此，如果b的某个元素存在误差，则解x的某些元素中的误差程度将不成比例. 矩阵条件数攀升至 10 13左右。
For n = 6, the number 4410000 is the element of maximum modulus of the matrix H − 1; but if we allow n to grow, the errors regarding the elements of x become ever greater. For n = 10, the maximum element, in modulus, of the inverse is 3.48 × 1012 and therefore, if there is an error present in some element of b, there would be a disproportionate degree of error in some element of the solution x. The matrix condition number climbs to around 1013.
所有这一切都与数学问题的结构有关，而不依赖于委托解决它的算法的选择。但是微积分系统对于算法的单个操作中的错误也有一定程度的敏感性。意识到克莱默求解十二个方程组的方法完全低效，威尔金森选择使用高斯算法，该算法包括将主对角线下的系数矩阵的所有元素减少到 0，从而将其减少为矩阵T三角形（对角线下方的元素等于 0）。Gauss 的方法比 Cramer 的方法效率更高，但它并不总是能够消除逐步计算的数字的异常增长。在这种增长上，微积分中的误差程度取决于，并且建立一个优于误差的极限作为三角矩阵元素大小的函数的公式以确定的精度表达了这种依赖性。
All of this has to do with the structure of the mathematical problem and does not depend on the choice of the algorithm that has been delegated to solve it. But there is also a degree of sensitivity of the system of calculus with regard to the errors made in single operations of the algorithm. Realizing that Cramer’s method for solving a system of twelve equations was completely inefficient, Wilkinson chose to use Gauss’s algorithm, which consists of reducing to 0 all the elements of the matrix of the coefficients under the principal diagonal, thus reducing it to a matrix T of triangular shape (with the elements beneath the diagonal equal to 0). Gauss’s method is incomparably more efficient than Cramer’s, but it is not always able to dispel an abnormal growth in the numbers that are progressively calculated. On this growth the degree of error in the calculus depends, and a formula that establishes a limit superior to the error as a function of the magnitude of the elements of the triangular matrix expresses this dependence with deterministic precision.
通常使用高斯方法产生的误差被限制在上端，表达式E由一个系数矩阵的大小的度量A乘以n的平方和一个增长因子。该因子由为三角矩阵T计算的元素的最大值与A的元素的最大值之间的关系（绝对值）精确定义。结果不具有概率意义，而是具有确定性意义，因为对于矩阵的维度n的每个可能值，误差永远不会大于E。9因此，我们可以得出结论，当 T的计算值 变得太大，整个计算图就失去了意义。误差放大的因素可以用多种方式表示，但最终总是取决于实际计算的数字的增长。
Usually the error made with the Gauss method is limited at the upper end with an expression E that consists of a measure of the magnitude of the matrix of coefficients A multiplied by the square of n and by a factor of growth. This factor is defined precisely by the relation (in absolute value) between the maximum of the elements calculated for the triangular matrix T and the maximum of the elements of A. The result does not have a probabilistic significance but a deterministic one, in the sense that for every possible value of the dimension n of the matrix, the error can never be greater than E.9 We may therefore conclude that when the calculated values of T become too big, the entire computational picture loses all sense. The factor of amplification of the error may be expressed in a variety of ways, but always ends up by depending on the growth of the numbers that have been actually calculated.
1946 年，Hotelling 确定了指数性质的误差极限，与 4 n成正比，这使得用自动程序以数值方式解决数学物理问题的希望渺茫。然而，最初的悲观预测将通过更准确的分析得到纠正。1947 年，John von Neumann 和 Herman Goldstine 发表了一篇历史性文章，他们在其中分析了正定对称矩阵求逆中的误差行为，10使用高斯方法，通过具有预定精度指标的算术。Von Neumann 和 Goldstine 证明了误差不会超过一个固定的量，但它会受到无法直接控制的波动的影响，这取决于矩阵的维数、执行计算的位数以及A的最大和最小特征值之间的关系。11它达到了一个可以说有点神奇的结果：对于大尺寸的矩阵，并且对于数百万或数十亿的近似运算，似乎有必要对舍入误差进行统计分析（无论如何都进行了）；但是冯诺依曼和戈德斯汀的结果具有确定性特征，即使在不确定区间的范围内：错误本身是不可知的，在任何情况下都不会超过某个阈值，该阈值至少在原则上是完美的能够被评估并依赖于计算数字的增长。
In 1946, Hotelling fixed a limit to error of an exponential nature, proportional to 4n, that left little hope for the possibility of resolving the problems of mathematical physics numerically with automatic procedures. Nevertheless, the initial pessimistic predictions were to be rectified by more accurate analyses. In 1947 John von Neumann and Herman Goldstine published a historic article in which they analysed the behaviour of the error in the inversion of a positive definite symmetrical matrix,10 with the Gauss method, by means of an arithmetic with a predetermined index of precision. Von Neumann and Goldstine demonstrated that the error, however subject to fluctuations that could not be controlled directly, does not exceed a fixed amount that depends on the dimensions of the matrix, on the number of digits with which the calculation is executed, and on the relation between the maximum and minimum eigenvalues of A.11 It amounted to a result that could be called somewhat miraculous: for large dimensions of the matrix, and for millions or billions of approximate operations, it could seem necessary to resort to a statistical analysis of the errors of rounding (which was in any case carried out); but the von Neumann and Goldstine result had a deterministic character, even though within the confines of an interval of indeterminacy: the error, in itself unknowable, would not in any case have gone over a certain threshold that was, at least in principle, perfectly capable of being evaluated and dependent on the growth of calculated numbers.
1971 年，威尔金森在一篇重要的文章中对过去 20 到 30 年间阐述的误差分析进行了调查，有人提示威尔金森评论说，断言在计算的某些阶段累积的误差在随后的阶段被放大是不准确的，并且事实上，如果操作的数量足够多，误差就会在统计上得到补偿——以至于自相矛盾地减少它们的影响。12最危险的敌人不是通过运算传播误差，而是矩阵求逆问题中固有的误差，即A的逆对微小变化的内在敏感性程度的元素A、无论采用何种方法。这种敏感性是通过条件指数来衡量的。
In 1971, in an important article surveying the analysis of error elaborated in the preceding twenty to thirty years, Wilkinson was prompted to comment that it was inaccurate to assert that the error accumulated in certain stages of the calculation is amplified in subsequent stages, and that in fact, if the number of operations is sufficiently high, the errors are compensated for statistically – so much so as to diminish, paradoxically, their effect.12 The most dangerous enemy is not the propagation of error through the operations, but the error inherent in the problem of the inversion of the matrix, that is to say, the degree of intrinsic sensitivity of the inverse of A with respect to the small variations of the elements of A, regardless of the choice of methods adopted. This sensitivity is measured by a condition index.
Alan Turing 在他 1948 年关于线性方程组的数值解和关于一般矩阵求逆的文章中首次明确了矩阵条件的概念，也就是说，不一定是对称的和正的确定的，元素等于实数。13图灵引入了一个范数，一个与矩阵相关的实数，一个旨在表达大小的索引 ——不是维度，由行数和列数给出，而是整体其元素的大小。图灵选择的范数，今天被称为Frobenius 范数，是由矩阵元素的平方和的平方根定义的：一个简单的数字，因此，测量整个矩阵的大小的任务被委托给它. 现在，图灵指出，可以将线性方程组解中出现的误差表示为系数矩阵A的元素中的误差乘以因子 μ 等于A的范数和A的倒数的范数（除以n ）。A元素中存在的误差的放大因此被认为取决于这个因素的大小，因此它具有条件数的含义。如果 μ 很高，我们将不知道计算出的数字是否会给出关于线性系统精确解的似是而非的信息，这相当于说我们将没有足够的关于方程解的信息，其中线性系统是一种算术近似，它定义了我们有兴趣预测的物理现象的数学模型。如果 μ 相对较小，则可以预测这种演变的进展。
The concept of the condition of a matrix was specified for the first time by Alan Turing in his 1948 article on the numerical solution of a system of linear equations and on the inversion of a general matrix, that is to say, not necessarily symmetrical and positive definite, with elements equal to real numbers.13 Turing introduced a norm, a real number associated with the matrix, an index that was meant to express the magnitude – not the dimension, given by the number n of rows and columns, but the overall magnitude of its elements. Turing’s chosen norm, today known as the Frobenius norm, was defined by the square root of the sum of the squares of the elements of the matrix: a simple number, therefore, to which the task of measuring the magnitude of an entire matrix was delegated. Now, Turing remarked, it is possible to express the error arising in the solution of a system of linear equations as the error in the elements of the matrix A of the coefficients multiplied by the factor μ equal to the product of the norm of A and the norm of the inverse of A (divided by n ). The amplification of the error that is present in the elements of A was therefore seen to depend on the magnitude of this factor, which as a result took the meaning of the condition number. If μ is high, we will not know if the calculated numbers will give plausible information on the exact solution of the linear system, which is equivalent to saying that we will not have sufficient information on the solution of the equations, of which the linear system is an arithmetical approximation that defines the mathematical model of the physical phenomenon the evolution of which we are interested in predicting. Predicting the progress of this evolution is instead possible if μ is relatively small.
这些考虑不依赖于选择求解线性方程组的算法：条件数仅取决于A的内在属性及其逆。然而，指标 μ 也干预算法的稳定性，即与数值求解方程组的实际过程有关，即误差通过运算传播的方式。计算结果对操作错误的敏感性的度量（必然近似）可以概括为一个基本上取决于 μ 的表达式。14这种情况已经在 von Neumann 和 Goldstine 1947 年关于对称和正定义矩阵求逆的文章中隐含。在本文中，误差不超过取决于因子 μ 的值，该因子可以表示为A 的最大和最小特征值之间的关系；但是这种关系与N ( A ) 和N(A -1 )之间的关系是一致的，其中N表示一个范数，欧几里得范数与图灵使用的 Frobenius 范数等价但不完全相同。15在 von Neumann 和 Goldstine 考虑的正定对称矩阵的特定情况下，欧几里得范数与矩阵的最大特征值一致。在这种情况下，同样的最大特征值在数值上测量矩阵的大小。
These considerations do not rely on the algorithm selected to solve the system of linear equations: the condition number depends only on the intrinsic properties of A and its inverse. Nevertheless, the index μ also intervenes in the stability of algorithms, namely, in relation to the actual procedures by means of which the system of equations is solved numerically, that is to say, the way in which the error propagates itself through the operations. The measure of the sensitivity of the result of the computation with regard to errors in the operations (necessarily approximated) may be summarized in an expression that depends essentially on μ.14 This circumstance was already implicit in the 1947 article by von Neumann and Goldstine on the inversion of symmetrical and positively defined matrices. In this article the error did not exceed a value dependent on a factor μ capable of being expressed as a relation between the maximum and minimum eigenvalues of A ; but this relation coincides with the relation between N(A ) and N(A−1), where N denotes a norm, the Euclidean norm that is equivalent but not identical to the Frobenius norm used by Turing.15 In the specific case of positive definite symmetrical matrices considered by von Neumann and Goldstine, the Euclidean norm coincides with the maximum eigenvalue of the matrix. In such a case, then, it is the same maximum eigenvalue that measures numerically the magnitude of the matrix.
总而言之，总体误差不超过取决于计算过程中数字增长方式的值。如果这些数字变得非常大，数学模型的可预测性的各个方面以及所有精度都会受到损害。这不仅是由于对非常大的数字进行四舍五入而导致的信息丢失或溢出风险的问题。关键的事实是计算数字的增长，增加了条件指数，阻碍了对错误行为的预测。该索引充当定义错误上限的表达式的基本组成部分，这是一个不能超过的限制。如果限制是一个很大的数字，错误可能很大，16正是这种不确定的情况使计算过程结束时打印的数字的含义变得令人怀疑并变得难以理解。
To conclude, the overall error does not exceed a value that depends on the way in which numbers grow during the computational process. If these numbers become very big, every aspect of predictability of the mathematical model is impaired, together with all precision. It is a question not merely of the loss of information due to the rounding of very large numbers, or of the risk of overflow. The crucial fact is that the growth of calculated numbers, increasing the condition index, impedes prediction of the error’s behaviour. This index acts as an essential component of the expression that defines the upper limit of error, a limit that cannot be exceeded. If the limit is a high number, the error might be big,16 and it is precisely this situation of uncertainty that puts into doubt and renders inscrutable the meaning of numbers printed at the end of the process of computation.
破坏的不仅仅是数字的增长计算系统；它们的减少也是如此，对于接近 0 的数字的后续操作会产生难以处理的结果，最终导致计算过程停止。在所有危急情况下，解决方案都会转瞬即逝，从某种意义上说是完全不真实的。有些定理在理论上假定它们的存在，但没有人能够计算出这些数字。
It is not just the growth of numbers that undermines a system of computation; their decrease does as well, and the consequent operations for numbers close to 0 can produce intractable results, ending with the arrest of the computation process. In all critical cases the solution becomes evanescent, in a sense completely unreal. There are theorems that posit, in theory, their existence, but nobody is in a position to be able to calculate the figures.
与康托尔和弗雷格所坚持的相反，可以肯定的是，数字并不都具有相同的本体论地位。存在但无法计算的数字与机器计算的数字具有不同的现实性。与后者不同，前者并不位于实际自动加工的空间和时间中——实际上不是或实际上不是。从某种角度来看，一个数字是存在的，是真实的，只有当有一个实际的程序来计算它时。但是这个过程也必须是有效 的：否则，就像在克莱默方法或希尔伯特矩阵的情况下，人们不知道如何在实际可实现性的水平上区分可计算的和不可计算的。可计算的类似于不可计算的。
What is certain, contrary to what Cantor and Frege maintained, is that numbers do not all share the same ontological status. Numbers that exist but cannot be calculated do not have the same reality as numbers calculated by a machine. The former, unlike the latter, are not situated in the space and time of an actual automatic elaboration – not actually or virtually in fact. From a certain point of view, a number exists, is real, only if there is an actual procedure that calculates it. But this procedure must also be efficient : otherwise, as in the case of Cramer’s method or Hilbert’s matrix, one would not know how to distinguish, on the level of actual realizability, between the calculable and the non-calculable. The calculable resembles that which is not calculable.
Boethius 解释说，无论数字如何彼此不同，数字都不是由它们本身以外的任何东西组成的。17数字域F的闭包概念将继续保证，通过将数字与F中定义的运算相结合，人们不会找到与它无关的元素，而总是并且只有F的元素。但是对于机器计算的数字，我们能说什么呢？在这种情况下，对数字的操作总是会生成其他数字；但是这些在计算过程中可能会失去所有意义。域的代数闭包不足以保证计算过程中生成的数字的重要性。
Boethius had explained that, however different from each other, numbers are not made up of anything other than themselves.17 The concept of the closure of a numerical field F would go on to guarantee that, by combining numbers with the operations defined in F, one does not find elements alien to it but always and only elements of F. But what can one say about numbers calculated by a machine? In this case, the operations on the numbers always generate other numbers; but these, during the course of computation, might lose all meaning. The algebraic closure of the field is not sufficient to guarantee the significance of numbers generated in a computational process.
整数的概念本身并不能将算法的概念解释为在时间和空间中展开的实际过程。Dedekind 曾寻求肯定的回应，但只是设法证明算术递归仅限于加法、乘法和除法运算，会导致回归到允许建立自然数概念的集合属性。
The concept of a whole number is not in itself capable of explaining in all its generality the concept of algorithm as an actual process that unfolds in time and space. Dedekind had sought an affirmative response, but had only managed to demonstrate that arithmetical recursion, limited to the operations of addition, multiplication and division, leads back to set properties that allow the concept of natural number to be founded.
我们所说的“有效”究竟是什么意思？该术语经常在文献中使用，但几乎从未以明确的方式定义。Andrej A. Markov Jr 试图解释算法的有效性是其基本特权之一，包括“算法获得特定结果的趋势”，18根据一组初始数据计算得出。因此，以递归方式定义的每个微积分过程都被称为实际的标准（等效于λ-formalism 或 Turing 的机器），理由是递归，早在 19 世纪的 Dedekind 理论中就已经是建设性地定义算术基本运算的技术。但在努力从理论上对其进行定义的同时，该算法也成为了面向解决应用科学问题的计算数学的主角。在这种背景下，至少从二十世纪下半叶开始，重点开始从有效性转向更加激进和可操作的效率需求。
What do we mean, exactly, by ‘effective’? The term is often used in the literature but is hardly ever defined in an explicit way. Andrej A. Markov Jr attempted to explain that the effectiveness of an algorithm is one of its essential prerogatives, consisting of the ‘tendency of the algorithm to obtain a certain result’,18 calculated on the basis of a set of initial data. As a result, it became standard to call actual every process of calculus defined in a recursive way (a criterion equivalent to λ-formalism or Turing’s machine), on the grounds that the recursion, already as early as Dedekind’s theory in the nineteenth century, is the technique for constructively defining the fundamental operations of arithmetic. But in parallel with the effort to define it theoretically, the algorithm also became the protagonist of a computational mathematics oriented towards resolving problems of applied science. In this context, at least from the second half of the twentieth century onwards, the accent began to shift from effectiveness to a more radical and operative demand for efficiency.
基本蓝图始终包括数学的算术化，将分析简化为整数的概念，以及在通往极限的过程中进行运算。但为了实现这种减少，需要有效的算法，既要建立复杂的计算策略并执行隐藏在这些策略中的基本算术计算，例如数字之间的加法和乘法，线性插值，多项式或矩阵的乘积。值得重复一遍：计算器执行数百万次操作，并使用大量不易访问的子程序。面对这种内在的、隐蔽的计算现象，人们必须能够依靠机器的算法，以尽可能最有效的方式准确地模拟该算法，这与稳定性和节省时间和空间。
The fundamental blueprint has always consisted of the arithmetization of mathematics, of the reduction of analysis to the concept of whole numbers and operations in the passage to the limit. But in order to realize this reduction, efficient algorithms are required, both to establish complex computational strategies and in order to execute the elementary arithmetical calculations hidden within these strategies, such as the additions and multiplications between numbers, the linear interpolations, the products of polynomials or matrices. It is worth repeating: the calculator executes millions of operations and makes use of numerous subprograms that are not readily accessible. Faced with the phenomenon of this immanent, concealed computation, one must be able to rely on an arithmetic of a machine that simulates that arithmetic exactly, in the most efficient way possible, in terms that relate both to stability and to the saving of time and space.
数学始终并且最重要的是追求经济原则：寻找获得预定目标的最短途径。这一原则不一定由有用的标准决定，它首先遵循一种理论简化的策略，该策略倾向于将最大的信息压缩到一个过程中，而花费在计算上的劳动力最少。如果我们想预测一个现象的行为，我们必须防止包含在数学模型中的信息被翻译成没有意义的算术碎片：在程序结束时获得的数字列表必须反映在较低的水平信息，在计算器的时间和空间中，模型打算表示的信息。数学在表示物理世界方面的功效是沿着理论模型和实际计算的数字之间的整个复杂而崎岖不平的中间道路传递的。在这篇文章的过程中，我们也可能失去与初始模型试图模拟的物理现象的性质的直接联系，因为计算的效率最终建立在纯数学之上，建立在与物理学无关的定理和数学结构上，但其效率产生的结果并不亚于理论模型与自然性质的初始对应关系。现象。在矩阵的情况下尤其如此：如果微分模型与线性方程组接近，则该系统的系数矩阵具有反映模型结构的结构，但解决方案可能涉及结构完全不同的矩阵。
Mathematics always and above all pursues a principle of economy: the search for the shortest route to obtaining a predetermined objective. This principle is not necessarily dictated by the criterion of what is useful and adheres above all to a strategy of theoretical simplification that tends to compress the maximum information into a process with minimum labour spent on calculation. If we want to predict the behaviour of a phenomenon, we must prevent the information contained within mathematical models from being translated into arithmetical detritus without meaning: the lists of numbers that are obtained at the end of the procedure must reflect, at a lower level of information, in the time and space of the calculator, that which the model intended to represent. The efficacy of mathematics in representing the physical world is transmitted along the whole complex and bumpy intermediate road between the theoretical model and the numbers that are actually calculated. In the course of this passage we can also lose direct contact with the nature of the physical phenomenon the evolution of which the initial model sought to simulate, because the efficiency of the calculation ends up ultimately being founded on pure mathematics, on theorems and mathematical structures that have little to do with physics, the efficacy of which nevertheless produces results that are no less miraculous than the initial correspondence of the theoretical model with the nature of the phenomenon. This obtains especially in the case of matrices: if, as usually happens, a differential model comes into proximity with a system of linear equations, the matrix of the coefficients of the system has a structure that reflects that of the model, but a solution may involve matrices of a completely different structure.
1946 年，赫尔曼·戈德斯汀 (Herman Goldstine) 和约翰·冯·诺依曼 (John von Neumann) 写了以下关于当时开始被描述的大规模数字计算的主题，特别是对通过微分模型解决流体动力学问题的数学解决方案：
In 1946 Herman Goldstine and John von Neumann wrote the following on the subject of the large-scale digital calculation that was beginning to be delineated at the time, with a particular bearing on the mathematical solution of problems of fluid dynamics by means of differential models:
在这一点上避免误解很重要：人们可能倾向于将这些问题归为物理学问题，而不是应用数学问题，甚至是纯数学问题。应该强调的是，这种解释是完全错误的。完全正确的是，所有这些现象对物理学家都很重要，并且通常主要受到他的赞赏。然而，这不应降低它们对数学家的重要性。事实上，我们认为，从纯数学的角度来看，应该赋予它们最大的意义。19
It is important to avoid a misunderstanding at this point: one may be tempted to qualify these problems as problems in physics, rather than in applied mathematics, or even pure mathematics. It should be emphasized that such an interpretation is wholly erroneous. It is perfectly true that all these phenomena are important to the physicist and are usually mainly appreciated by him. Yet, this should not detract from their importance to the mathematician. Indeed, we believe that one should ascribe to them the greatest significance from the purely mathematical point of view.19
然而，微积分必须被理解到最后的细节。哲学所提供的伟大的理论综合体也寻求符合一种现实原理，与最基本的计算操作精确相关。从这个意义上说，数学被认为是还原论的，但还原论不能鼓励我们忘记，模型和实际计算的数字之间的所有中间阶段都是可能的，这要归功于理论概念和相对抽象的数学结构的介入。在对现实的扩展概念中，最抽象的结构与数字列表一样重要——这本身是难以理解的——计算器在计算过程结束时打印出来：它们是彼此的镜像，每个被对方解密。因此，数学不仅仅是一种抽象。但正是由于其抽象性和理论内容，
Nevertheless, calculus must be understood down to its last details. The great theoretical syntheses provided by philosophy have also sought, in compliance with a principle of reality, a precise correlation with the most elementary operations of computing. In this sense, mathematics is held to be reductionist, but the reductionism must not encourage us to forget that all the intermediate stages between the model and the numbers actually calculated are possible only thanks to the intervention of theoretical concepts and relatively abstract mathematical structures. In an expanded idea of reality the most abstract structure counts just as much as the list of numbers – which by itself is unintelligible – that a calculator prints materially at the end of a process of calculation: they are mirror images of each other, with each being decrypted by means of the other. Therefore, mathematics is not just an abstraction. But precisely because of its abstract nature and its theoretical content, it is a foundation of the reality of this world and of the way in which we venture to intervene in order to modify it.
数学与我们准备将其认定为真实和实际的事物的接近性唤起了希腊人分配给最简单操作的那种力量的性质，例如在一条线上构建一个正方形，柏拉图用术语dýnamis会签了它。那时，效力（dýnamis ）已经能够生产，除了神殿和祭坛的增长，最复杂的机械装置、弹射器和其他战争机器。因此，柏拉图式的动态是不可避免的可以将自己转化为实际的科学和技术力量，并承担我们现在可以想象和抵消的所有风险。在斯宾诺莎的《伦理学》中可以找到一个明确的现实与权力之间密切关系的标志，这种亲和性可以扩展到整个自然范围。它的行动力无处不在。结论是：“任何事物的力量或力量……只不过是事物本身的既定本质，即实际本质。” （伦理学，III，序言和命题 7，示范。）
The proximity of mathematics to that which we are prepared to qualify as real and actual evokes the nature of that power which the Greeks assigned to the simplest operations, such as constructing a square above a line, and which Plato has countersigned with the term dýnamis. Already back then potency (dýnamis ) was capable of producing, besides the growth of temples and altars, the most sophisticated mechanical contraptions, catapults and other war machines. It was therefore inevitable that the Platonic dýnamis could transmute itself into actual scientific and technical power, with all the risks that we can now imagine and counteract. A clear sign of the affinity between reality and power, which may be extended to the entire scope of nature, is to be found in Spinoza’s Ethics, in a period already profoundly marked by the Promethean leap of science: ‘nature is always the same, and its power to act is everywhere.’ And in conclusion: ‘The power or force of anything … is nothing but the given, that is to say actual, essence of the thing itself.’ (Ethics, III, Preface and Proposition 7, Demonstration.)
17. 基本面的危机和复杂性的增长：现实与效率
17. The Crisis of Fundamentals and the Growth of Complexity: Reality and Efficiency
不仅仅是数字在增长，微积分的复杂性也在增加。而这个阴险的特征也可能威胁到最知名和最受考验的程序。受影响的不仅是 Cramer 的算法。如果我们使用经典的高斯方法来求解线性方程组或计算矩阵的行列式，1计算量的增加可能会导致效率上的缺陷：在行列式的情况下，算术运算是维数为n的多项式函数，但并不总是容易验证二进制数操作也仍然是多项式的，对于在计算中干预的数字的以 2 为底的表示的单个和的基本运算。这个数字可能会变成指数。2
It is not just the numbers that grow, the complexity of calculus also increases. And this insidious feature may also threaten the best-known and most-tested procedures. It is not only Cramer’s algorithm that is affected. If we employ the classic Gauss method to resolve a system of linear equations or to calculate the determinant of a matrix,1 a flaw in efficiency may arise from the increase of the size of the calculations: in the case of the determinant, the number of arithmetical operations is a polynomial function of the dimension n, but it is not always easy to verify that the number of binary operations also remains polynomial as regards the elementary operations on the single sums of the representations in base 2 of the numbers that intervene in the calculation. This number may become exponential.2
计算复杂性的研究，即衡量计算函数难度的标准，一直是 50 年代以来计算机科学最重要的方向之一。随着维度的增加，由于无法控制的组合爆炸，解决相对初级问题所带来的难度会以指数方式增长。所谓的 NP 完全问题就是这种情况（可在多项式时间内使用非确定性机器求解）：验证给定解决方案是否针对这些问题中的任何一个实际上都是如此，但是根据目前的知识，实际寻找解决方案需要一些至少是指数级的操作。
The study of computational complexity, that is to say, of the criteria for measuring how difficult it is to calculate a function, has been one of the most important streams of computer science since the fifties. With the increase in dimension, the difficulty presented by the resolving of relatively elementary problems can grow in an exponential way, due to an uncontrollable combinatorial explosion. This is the case with the so-called NP-complete problems (solvable in Polynomial time with a Non-deterministic machine): it is relatively easy to verify if a given solution proposed for any one of these problems is actually the case, but the actual search for a solution requires, according to current knowledge, a number of operations that is at least exponential.
复杂性理论也助长了威胁 20 世纪头几十年数学的基本面危机，为在 40 年代之前被完全忽视的领域的研究开辟了新的视角。由于所解决的问题之间以及澄清这些问题所必需的技术之间的相似性，这种重估得以发展：启动基础研究的核心问题，来自希尔伯特在 23 年提出的著名问题清单1900 年的巴黎，关于是否可以解决任何数学问题，无论是积极的意义（通过实际计算解决方案）还是消极的意义（通过证明不存在可以计算解决方案的算法）。与微积分的复杂性有关的核心问题是数学问题是否可以在多项式时间内解决，也就是说，是否可以使用等于问题大小的多项式函数的运算数量，从而在最终分析，用计算器记忆的时间和空间。
The theory of complexity also contributed to the galvanizing of the crisis of fundamentals that had threatened the mathematics of the first decades of the twentieth century, opening new perspectives for research in areas that had been completely ignored until the forties. This revaluation was able to develop thanks to the affinities between the problems tackled, as well as between the techniques that were necessary to clarify them: the central question that initiated research into fundamentals, from the celebrated list of twenty-three problems posed by Hilbert in Paris in 1900, pertained to whether any mathematical problem can be solved, whether in a positive sense (with the actual calculation of the solution) or in a negative one ( by demonstrating that no algorithm exists which could calculate the solution). The central question relating to the complexity of calculus is whether a mathematical problem can be resolved in polynomial time, that is to say, with a number of operations equal to a polynomial function of the size of the problem, and hence compatible, in the final analysis, with the time and space of the calculator’s memory.
在 1956 年普林斯顿大学的一封信中，哥德尔向冯诺依曼提出了一个数学问题，该问题注定会成为微积分科学的一个基本问题（但仍未解决）：哥德尔认为，我们可以很容易地构建一个图灵机，对于一阶谓词计算的每个公式F和每个固定自然数n，我们可以判断是否存在长度为 n 的F 的演示（如果长度n是符号数，则只需指定列表控制机可以执行的所有长度为n的演示，然后检查其中是否有F 的演示）。现在，如果p是机器为此目标所需的步数（取决于F和n ），并且如果P ( n ) 是变化F时数字p之间的最大值，那么 P ( n ) 增长的速度有多快 ，当n增加时，对于最优机器？
In a letter from Princeton dated 1956, Gödel put to von Neumann a mathematical question that was destined to become one of the fundamental (and still unresolved) problems of the science of calculus: we may easily construct, argued Gödel, a Turing machine that, for every formula F of the calculation of the predicates of the first order and for every fixed natural number n, allows us to decide whether there is a demonstration of F of length n (if the length n is the number of symbols, one need only specify the list of all the demonstrations of length n that can be executed by the controlling machine, then check if among these there is the demonstration of F ). Now, if p is the number of steps (depending on F and n ) that the machine requires for this objective, and if P (n ) is the maximum between the numbers p when varying F, how quickly does P (n ) grow, when n increases, for an optimal machine?
因此，哥德尔提出了一个问题，该问题不仅涉及纯算法的可解性，还涉及计算复杂性，以及计算难度的测量。Juris Hartmanis 对这封信的评论有助于更好地阐明其在数学基础研究领域中的战略重要性。Hartmanis 解释说，从哥德尔的结果来看，我们知道在数学中存在如此丰富的公式和问题，以至于它（数学）无法在连贯和完整的意义上进行公理化解决。从图灵和丘奇的结果中，我们还了解到，对于足够复杂的形式系统，哪些断言可证明或不可证明的一般问题不是可以通过算法过程来决定的。哥德尔在他的信中提出了下一个问题，微积分复杂性的增长 ：随着 n 的增加，用一个过程来决定在一个正式的系统中，一个断言是否允许一个长度为n的证明有多困难？3
Gödel thus posed a question that did not concern pure algorithmic solvability alone but also computational complexity, as well as the measurement of the difficulty of the calculation. A comment on the letter made by Juris Hartmanis helps to better clarify its strategic importance within the field of research on the foundations of mathematics. From Gödel’s results, Hartmanis explains, we know that within mathematics there is such an abundance of formulas and problems that it (mathematics) is incapable of axiomatic resolution in a coherent and complete sense. From Turing and Church’s results we also learn that for formal systems that are sufficiently complex the general question of which assertions are or are not demonstrable is not something that may be decided through an algorithmic process. In his letter Gödel then raised the next question, which still has to do with the foundations of mathematics, but now in terms of the growth in complexity of calculus : with the increase of n, how difficult is it to decide with a procedure whether, in a formal system, an assertion admits of a demonstration of length n?3
哥德尔摧毁了公理化数学的梦想，为了具有讽刺意味的效果，他现在冒险提出更乐观的假设，即冯诺依曼提出的问题的复杂性可能是线性的，或者最多是二次的。我们现在知道这属于 NP
完全问题的类别。这门课的学习要求技术完全类似于用于纯理论可解性问题的技术，这使得有可能（否定地）回答希尔伯特在他著名的 1900 年巴黎演讲中提出的一些问题。
Gödel, who had destroyed the dream of an axiomatized mathematics, was now hazarding, for ironic effect, the more optimistic hypothesis that the complexity of the problem posed by von Neumann could be linear or, at most, quadratic. We now know that this belongs to the class of NP-complete problems. The study of this class has required techniques altogether akin to those used for questions of pure theoretical solvability which had made it possible to reply (negatively) also to some of the queries raised by Hilbert in his celebrated 1900 lecture in Paris.
关于计算复杂性的更多理论问题将在几个方面与负责解决各种应用问题的数值算法的效率度量交织在一起。维度的增长方式、操作的数量、实际计算的数量或算法错误或由于病态导致的增长方式在一种或另一种情况下是确定（如果不是现实）实际存在的标准解决方案。但是这些新问题，即使是根本性的并且部分未解决，也没有引发与 20 年代影响数学的危机相媲美的危机。现在的紧迫性似乎已经转移到与理论计算机科学和计算数学应用有关的问题上。
The more theoretical questions on computational complexity would be intertwined, in several respects, with the measure of efficiency of the numerical algorithms entrusted with the task of resolving various kinds of applied problems. The modalities of growth of the dimensions, of the number of operations, of the numbers actually calculated or of algorithmic error or what is due to ill conditioning are in one case or another a criterion for establishing, if not reality, then the actual existence of solutions. But the new questions, even though fundamental and partly unresolved, did not provoke a crisis comparable to that which affected mathematics in the twenties. The urgency seems now to have shifted to problems relating to theoretical computer science and the application of computational mathematics; to a comprehensive science of calculus that is in itself sufficiently empirical and flexible to tolerate operational imperfections or uncertainties.
算法的效率首先取决于两个因素：它的复杂性和稳定性，也就是说，它对操作错误的整体敏感性。谈论一个数学问题的解决方案是合理的和不可避免的，但在大多数情况下，这个解决方案根本不为人所知，并且其解析公式的计算（应该存在）将导致近似误差大于那些从用更简单的模型替换初始模型得出。通过用算术模型代替微分或积分模型，然后仅使用对数字的基本运算来计算解：加法、减法、乘法阳离子和分裂。然而，可能提供的知识总是近似的，计算结果可能完全不可行。因此，总而言之，数字的基本算术所固有的确定性程度，希尔伯特构建他的（元数学）技术的模型，旨在证明无限的数学用途是合理的，并不是完全站得住脚的。即使是用计算器执行的一次加法，也可以生成没有意义的数字。4
The efficiency of an algorithm depends above all on two elements: its complexity and its stability, which is to say, its overall sensitivity with regard to errors in operations. To speak of the solution of a mathematical problem is justified and inevitable, but in the majority of cases this solution is not known at all and the calculation of its analytical formula, which is supposed to exist, would entail errors of approximation greater than those that follow from the substitution of the initial model by a simpler one. By substituting the differential or integral model with an arithmetical model the solution is then calculated with only the fundamental operations on numbers: additions, subtractions, multiplications and divisions. Nevertheless, the knowledge that may be afforded is always approximate, and the calculated results may be completely unviable. So, in conclusion, the degree of certainty intrinsic to the elementary arithmetic of numbers, on the model of which Hilbert constructed his (metamathematical) techniques designed to justify the mathematical use of the infinite, was not entirely defensible. Even one single addition, executed with a calculator, can generate numbers devoid of meaning.4
误差类型学与迭代类型的计算过程有关：通过重复应用运算符，生成一系列数字，其与解的距离（可以解释为误差）趋于 0。从类似的计算过程中，推导了序列的极限和收敛的分析概念，收敛的某些证明仍然基于相同计算过程的存在和性质，而不是基于逻辑论证。但是，我们是否真的能够通过迭代过程，以将（误差）差距缩小到任意小的值的方式越来越接近解决方案？数学家们花了很长时间来问自己这个问题，今天，用柯西使用的同样的话，任意关闭”。5然而，这种不确定的解决方法仅在理论上有效，并且近似误差不能变得任意小。四舍五入的误差在方程的根周围产生了致命的不确定性区间，该区间使计算出的所有意义的数值超出了一定的近似极限：一个好的算法可能会提供，经过一定的步数，该间隔内的一个值，但通过计算同一间隔内的另一个连续步骤来改进该值是没有意义的。
A typology of error pertains to computational processes of an iterative type: with repeated applications of an operator one generates a succession of numbers the distance of which from the solution (which can be interpreted as error) tends towards 0. From similar processes of calculation, the analytical concepts of the limit and the convergence of a sequence were derived, and certain demonstrations of convergence are still based on the existence and properties of this same processes of calculation rather than on logical arguments. But are we actually capable, with an iterative procedure, of getting ever nearer to a solution in a way that closes the gap (of error) to an arbitrarily small value? The mathematicians took a long time getting round to asking themselves this question, and today, in the same words as used by Cauchy, we can still hear how it is possible to calculate for the root of an algebraic equation ‘approximate numerical values that are arbitrarily close’.5 Nevertheless, this indefinite approach towards the solution is valid only in theory, and the error of approximation cannot become arbitrarily small. The errors of rounding create, fatally, around the root of the equation, an interval of uncertainty that divests the calculated numerical values of all meaning beyond a certain limit of approximation: a good algorithm may furnish, after a certain number of steps, a value within that interval, but it makes no sense to look to improve that value by calculating another, successive one within the same interval.
如前所述，为了解释连续统由什么组成，庞加莱要求我们考虑两条相交的线。我们可以想象它们是在笛卡尔平面上绘制的。我们的想象力倾向于将它们想象成两条细丝带，它们在相交时必须具有共同的一部分。数学家更进一步：在不拒绝想象的建议的情况下，他试图构想一条没有宽度的线和一个没有延伸的点。然后他想到这条线是一个界限色带随着宽度逐渐减小而趋向于该位置。两条带之间的交叉区域随着它们的宽度向 0 移动而趋于的极限是一个点。这就是为什么，Poincaré 总结说，我们倾向于根据一个直觉的真理，坚持两条相交的线必须有一个共同点。
As already mentioned, in order to explain what the continuum consists of, Poincaré asked us to consider two intersecting lines. We can imagine them drawn on the Cartesian plane. Our imagination tends to visualize them as two thin ribbons which, on intersecting, are obliged to have a part in common. The mathematician takes a further step: without rejecting the suggestion of the imagination, he tries to conceive of a line without width and a point without extension. It then occurs to him to think of the line as a limit to which a ribbon tends as it is progressively reduced in width. The limit to which the area of intersection between the two ribbons tends as their width moves towards 0 is then a point. This is why, Poincaré concluded, we are inclined to maintain, in compliance with an intuitive truth, that two intersecting lines must have a point in common.
如果曲线的交点P的横坐标α不对应有理数会怎样？使用实数域的理论定义所建议的策略，我们尝试计算一系列长度递减的区间，其中包括α，每个区间都包含在前一个区间内。理论上，我们有权假设我们可以任意定义一个包含α的长度区间小的。但是，如果我们使用最知名的数值算法来识别这些区间，一步一步地，我们就会意识到不可能将它们的长度减少到超过一定限度。这种不可能的原因在于需要停止或四舍五入实际计算的数字序列。
What happens if the abscissa α of the point of intersection P of the curves does not correspond to a rational number? With the strategy suggested by the theoretical definitions of the field of real numbers we try to calculate a sequence of intervals of decreasing length that include α, each contained within the preceding one. In theory, we are entitled to assume that we can define an interval of length containing α that is arbitrarily small. But if we use the best-known numerical algorithms designed to identify these intervals, step by step, we realize that it is not possible to reduce their length beyond a certain limit. The reason for this impossibility resides in the need to arrest or round off the sequences of numbers that have actually been calculated.
如果两条曲线在笛卡尔平面上由两个函数f ( x ) 和g ( x ) 定义，则必须数值求解的方程为h ( x ) = 0，其中函数h由差定义，对于每个x，在f ( x ) 和g ( x ) 之间，即h ( x ) = f ( x ) - g ( x )。数α的理论存在取消函数h由著名的 Bernhard Bolzano 定理保证：如果在极端a和b的闭区间 [ a , b ] 中，也就是说，一组实数x其中a ≤ x ≤ b，一个连续函数，对于x的某些值具有正号，而对于其他值具有负号，则在a和b之间包含一个数字α，该函数被取消（在图 9中，函数h在a中具有正号和b 中的负数）。计算α的策略经常求助于这个定理：计算极值区间越来越近，因此函数h在它们中的每一个中改变符号。这些区间的不定序列定义了数α，因此原则上存在于实数体中。但是，对于近似于α的数字的实际计算，我们能说什么呢？这种计算通常意味着在不同点对函数h的评估，以便每次确定它是正的还是负的。但由于四舍五入的误差，我们不计算h ( x )，而是一个函数t ( x ) = h ( x ) + e ( x )，其中e ( x ) 是不超过某个正数ε的扰动。现在如果 [ c , d ] 是围绕α的最大圆形区间，其中函数h的绝对值小于或等于ε，不难证明如果t ( x ) 在那个区间，对于h不一定如此( x ) 也是如此。6
If the two curves are defined on the Cartesian plane by two functions f (x ) and g(x ), the equation that must be resolved numerically is h(x ) = 0, where the function h is defined by the difference, for every x, between f (x ) and g(x ), that is to say, h(x ) = f (x ) - g(x ). The theoretical existence of the number α which cancels the function h is guaranteed by the celebrated theorem of Bernhard Bolzano: if in a closed interval [a, b ] of extremes a and b, that is to say, a set of real numbers x for which a ≤ x ≤ b, a continuous function that has a positive sign for some values of x and a negative one for others, then there is a number α included between a and b for which the function is annulled (in Fig. 9 the function h has positive sign in a and a negative in b ). The strategies for calculating α frequently resort to this theorem: one calculates extreme intervals ever closer, so that the function h changes sign in each of them. The indefinite sequence of these intervals defines the number α, which exists thus, in principle, in the body of real numbers. But what can one say of the actual calculation of numbers with which one approximates α? This calculation implies, typically, the evaluation of the function h at different points, in order to determine each time if it is positive or negative. But due to errors of rounding we do not calculate h(x ), but rather a function t(x ) = h(x ) + e(x ), where e(x ) is a disturbance that does not exceed a certain positive number ε. Now if [c, d ] is the maximum circular interval around α, inside which the function h is in absolute value less than or equal to ε, it is not difficult to demonstrate that if t(x ) is positive (or negative) in that interval, it is not necessarily so for h(x) as well.6
从计算x落入区间 [ c , d ] 的那一刻起，我们验证函数h是否采用不同符号的计算不会得出任何结论。这意味着不可能计算出一个小于 [ c , d ] 的区间，其中包含方程h ( x ) = 0 的解α，即两条曲线交点的横坐标。x在最终无法减少的区间内的位置将被忽略。
From the moment in which x is calculated to fall in the interval [c, d ], our calculations to verify whether that function h assumes different signs do not lead to any conclusion. This implies that it is not possible to calculate an interval smaller than [c, d ] that contains within it the solution α of the equation h(x ) = 0, that is to say, the abscissa of the intersection of the two curves. The location of x within an interval that cannot ultimately be reduced is ignored.
图 9
Figure 9
类似的推理自然适用于比所描述的更复杂的问题，例如非线性方程组的解或涉及最小值的问题。因此，经常围绕着一个问题的解决方案的不确定性的云，使得它的存在本身就有问题。我们所知道的关于该问题的所有信息都在于指定用于近似它的算法，并且可以理解，该算法必须能够提供有关适用于其计算目标的解决方案的信息。从某种意义上说，解决方案就是算法，但这也需要效率算法的稳定性（对错误的敏感性）和计算复杂度。因此，在建立数字本体时，算法的可构造性和效率都是必要的。效率是计算数学和大规模自动计算的标准，它完善了在理论可计算性背景下引入的有效性的纯粹概念。
Similar reasoning is naturally valid for more complex problems than the one described, such as the solution of a system of non-linear equations or a problem involving the minimum. Hence the cloud of indetermination that regularly surrounds the solution of a problem renders problematic its very existence. All that we know about that problem lies in the algorithm designated to approximate it, and it is understood that this algorithm must be capable of giving information about the solution adapted to the objectives for which it is being calculated. In a sense, the solution is the algorithm, but this also requires the efficiency of the algorithm, measured in terms of stability (sensitivity to errors) and computational complexity. Hence algorithmic constructability and efficiency are both requisite when establishing an ontology of numbers. Efficiency is a criterion of computational mathematics and of large-scale automatic calculation that completes the pure idea of effectiveness introduced in the context of theoretical computability.
这个结论似乎将数字的现实从复杂的理论定义转变为具体算法效率的概念。也就是说，为了使算法有效，需要将数学实体插入到最具体的计算中，这些数学实体可以参考定理和相对抽象的数学结构，并且通常远离旨在模拟现象的模型。例如，干预过程的矩阵——其唯一目的是简化计算——通常具有特殊的结构，在某些情况下，可以通过代数群的性质来抽象地描述这种结构。只是因为它需要具有结构的矩阵，而不是通用矩阵，是否有可能得出否则不可能得出结论的计算。因此，效率的重要性以数学实体的存在为前提。正如已经指出的那样，Goldstine 和 von Neumann 已经警告说，数字微积分在很大程度上是基于纯数学的。
This conclusion seems to shift the reality of numbers from a complex of theoretical definitions to an idea of concrete algorithmic efficiency. That said, for an algorithm to be efficient one needs to insert into the most concrete calculation mathematical entities which may be referred back to theorems and to mathematical structures that are relatively abstract and often distant from the model with which one aims to simulate a phenomenon. For example, the matrices that intervene in a procedure – with the sole aim of simplifying calculations – usually have a special structure, which may be described in abstract terms, in certain cases, by means of the properties of an algebraic group. Only because it entails matrices with structure, and not generic matrices, is it possible to bring to a conclusion calculations that would otherwise be impossible. Therefore, the materiality of efficiency presupposes the existence of mathematical entities. Goldstine and von Neumann, as has been pointed out, had already warned that digital calculus is in large part based upon pure mathematics.
过去已经提出的论点，即只要以连贯的方式定义数字，就具有相同的本体论地位，不再是合理的——但同样的情况是，它们的具体存在取决于建立其可计算性的条件和限制。
The thesis, already advanced in the past, that numbers have the same ontological status as long as they are defined in a coherent way, is no longer plausible – but it is equally the case that their concrete existence depends on a theoretical knowledge that establishes the conditions and limits of their calculability.
18. Verum et Factum
18. Verum et Factum
Verum et factum convertuntur（“真实和创造是可以互换的”）：这是詹巴蒂斯塔·维科在拉丁文学、特伦斯和普劳图斯中发现的重要证据，今天仍然可以作为建构主义认识论的前提。某事的真实性取决于“做”，取决于实际将其与行动相结合。因此，数学问题的解决取决于在自动执行的物理空间和时间中以有效方式计算它的可能性，这是给定问题规模的唯一可能的策略。似乎没有什么比一个程序更确定的了，该程序通过有限数量的步骤，对分配的数据进行必要的计算。但是，数学实体的现实是否真的以一种全面的方式被封装，
Verum et factum convertuntur (‘the true and the created are interchangeable’): this is the formula that Giambattista Vico found significant evidence of in Latin literature, in Terence and in Plautus, and which today can still serve as a premise for constructivist epistemologies. The reality of something depends on the ‘doing’, on actually bringing it to term with an action. The solution of a mathematical problem depends, then, on the possibility of calculating it in an efficient manner in the physical space and time of an automatic execution, which is the only strategy possible, given the size of the problems. There seems to be nothing more certain than a procedure which, through a finite number of steps, carries out the necessary calculations in relation to assigned data. But is the reality of mathematical entities really encapsulated, in a comprehensive way, in this conclusion?
有必要记住，所有分析及其作为描述物理世界的模型的功能都是基于 Cantor 和 Dedekind 定义的实数域的度量属性。这个以算术方式表达连续性概念的数字领域，在其独特的品质中具有完整性。这意味着任何基本的实数序列都收敛于一个实数，也就是说，它承认一个不在实数域之外但属于实数域的极限。有理数的领域并不完整，因为有一系列有理数允许有一个非有理的极限——而这是横向和对角数的情况，它们的关系收敛于
It is necessary to remember that all analysis and its function as a model for the description of the physical world are based on the metric properties of the field of real numbers defined by Cantor and Dedekind. This field of numbers, which expresses arithmetically the concept of continuity, has among its peculiar qualities that of completeness. This means that any fundamental sequence of real numbers converges with a real number, that is to say, it admits a limit that is not outside but belongs to the field of real numbers. The field of rational numbers is not complete, because there are series of rational numbers that allow a limit that is not rational – and this is the case for lateral and diagonal numbers, the relations of which converge with 2, which is not rational. Now, one may frequently demonstrate the existence of a solution to a mathematical problem, such as an algebraic equation, an integral or a differential equation, by having recourse to the property of completeness: one calculates a sequence of values that approximate the solution and demonstrates that this sequence is fundamental. For the property of completeness, the limit of this sequence exists in the field of real numbers.
到目前为止，我们只是承认解的存在，但为了做到这一点，我们使用了一系列近似它并且实际上是可计算的值。解决方案存在的证明求助于计算逐次逼近的算法。因此，描述算术连续统的完备性理论特性以及其他特性，被应用于由微分模型描述的物理问题，人们寻求通过必须满足精确收敛特性的算法来解决这些问题。由于问题的维度增加，计算的执行必须是自动的，直到对无穷无尽的数字列表进行最终评估。这些列表，就像设计用来计算它们的程序一样，本身可能无法理解；然而，它们包含，
Up to this point we merely admit the existence of the solution, but in order to do so we use a sequence of values that approximate it and that are actually calculable. The demonstration of the existence of the solution resorts to an algorithm that calculates the successive approximations. Hence the theoretical property of completeness that characterizes, along with others, the arithmetical continuum, is applied to physical problems described by differential models which one seeks the solution to with algorithms that must satisfy a precise property of convergence. The execution of the calculations must be automatic, due to the elevated dimension of the problems, up to the final evaluation of interminable lists of numbers. These lists, like the procedures designed to calculate them, may not by themselves be intelligible; nevertheless, they enclose, in terms of pure sequences of digits, the information communicated by the model and hence by the same physical phenomenon which the model is seeking to simulate.
计算本身最终变成了一个物理过程，但很难将其与使之成为可能的纯数学理论假设分开。问题解决方案的实际存在当然取决于算法的效率，但如果人们试图掌握数学实体的真实性，那就是很难将相交和相互依赖的推理和计算领域分开，直到它们形成一个完全成熟的理论——定理、论证和算法，在此基础上，这种现实得到全面支持和几乎强加。正如西蒙娜·威尔（Simone Weil）所写，“真实是强加于自己的东西。演示比感觉更能强加给我们。但它部分源于惯例。有必要在数学中捕捉非常规。1事实上，在从数学模型到数字微积分的过程中，几乎没有什么是传统的：物理事件的性质必然会在方程的结构和旨在解决它们的数学实体中传递，直到它被印上在最后计算的数字列表中。归根结底，被定义为“数学在自然科学中的不合理有效性” 2似乎是属性和环境的复杂而清晰的组合的结果，这些组合显着削弱了人们对两者之间可能联系的偶然特征的初步印象。数学和物理世界的抽象概念。
The calculation itself, ultimately, becomes a physical process, but it is difficult to separate it from the theoretical presuppositions, purely mathematical, that render it possible. The actual existence of the solution of a problem is certainly predicated on the efficiency of an algorithm, but if one attempts to grasp the reality of the mathematical entity it is difficult to separate the domains of reasoning and calculation that intersect and rely on each other, until they form a fully fledged theory – of theorems, demonstrations and algorithms, on the basis of which such reality is comprehensively supported and almost imposed. As Simone Weil writes, ‘the real is that which imposes itself. The demonstration imposes itself on us more than the sensation. But it partly derives from convention. It is necessary in mathematics to capture the non-conventional.’1 In fact, in the procedure that leads from the mathematical model to digital calculus there is little that is conventional: the nature of the physical event is necessarily transmitted in the structure of the equations and the mathematical entities designed to resolve them, until it is imprinted in the last lists of numbers calculated. Ultimately, that which has been defined as ‘the unreasonable effectiveness of mathematics in the natural sciences’2 appears to be the consequence of a complex and articulated combination of properties and circumstances that significantly attenuate the initial impression of the accidental character of the possible connections between the abstract concepts of mathematics and the physical world.
事实位于建模过程的最后，体现在实际计算的数字中，并位于计算器的内存中；尽管如此，它也是允许构建和研究模型的算术连续统理论的一个前提——历史的和概念的——。截面的概念和戴德金的算术连续统是在算法结构中表达的一个原始事实的自然结果：“分区的概念[在戴德金的术语中的截面或切割]之前是否有一个纯粹的算法事实，也就是说，通过需要证明某些算法程序的合理性和合法性，例如通过过度和缺陷的近似
The factum is situated at the end of the process of modelling and materializes in the numbers actually calculated and located in the memory of the calculator; nevertheless, it is also a presupposition – historical and conceptual – of the theory of the arithmetical continuum that allows the construction and study of the model. The concept of section and Dedekind’s arithmetical continuum are the natural consequence of a primordial fact that is expressed in an algorithmic construction: ‘Is it not perhaps true that the concept of partition [of section or cut, in Dedekind’s terms], is preceded by a purely algorithmic fact, that is to say, by the need to justify and legitimize certain algorithmic procedures, such as the approximation by excess and defect of 2, that are precisely translated into the construction of the classes made up of infinite discrete numbers?’3
可以在一定程度上断言verum et factum convertuntur , true 最终可以与 made 互换 - 但我们还必须考虑到很难对“made”做出精确定义的事实，或应该构成计算过程本质的“有效性”。此外，坚持“现实的境界无限超越创造的境界”的论点4当然不是不可接受的，这取决于可能以某种方式而不是以另一种方式发生的情况的组合。因此，制造商对其基本原理的研究仍然漠不关心。正是后者作为一种积极的和生产性的原则，实际上包含了实践中存在的东西。正如托马斯·阿奎那所解释的那样，在某物潜在地存在之前，必须有某物在实践中存在，因为如果不是通过实践中已经存在的某物，潜力不会在实践中自行解决。5
It is possible to assert, to a certain degree, that verum et factum convertuntur, the true is ultimately interchangeable with the made – but we must also take into account the fact that it is difficult to make a precise definition of ‘the made’, or of the ‘effectiveness’ that should constitute the essence of a computational process. Moreover, the thesis that maintains that ‘the realm of reality infinitely surpasses that of the made’4 is certainly not inadmissible, depending on a combination of circumstances for which the made may occur in a certain way and not in another. The made, as such, remains indifferent with regard to research into its fundamentals. And it is the latter that virtually contains, as an active and productive principle, what exists in practice. As Thomas Aquinas explained, before something exists potentially there must be something that exists in practice, because potentiality does not resolve itself in practice if not by means of something that already exists in practice.5
19. 递归和不变性
19. Recursion and Invariability
增长现象不是一个边缘问题，因为它涉及到算法最亲密的结构。增长方式在决定效率方面具有决定性作用。当解释算法以递归方式运行时，其含义或多或少是：算法旨在解决的问题（例如，每个十六位数的两个数字的乘积）被划分成相同类型但维度较小的问题（八位数字的乘积），然后将这些问题依次划分为更小的类似问题（四位数字的乘积）。除法过程继续进行，直到遇到最小维度的基本问题（只有一位数字的乘积）。从后者，通过迭代演算，一个人在相反的方向上追溯，直到一个人解决了最初提出的问题。这种组织递归演算的方法，通过相同类型但降维的问题的层次结构，称为divide et impera（“分而治之”），来自古老的拉丁格言。但划分标准本身并不总是且必然有效。重要的是调节维度减少和连续增长的规律。Cramer 的方法将 n 维矩阵的行列式的计算追溯到n - 1维矩阵的行列式的计算，是完全低效的。各种各样的算法使用取而代之的是二分法，也就是说，将最初的n维问题分成n /2维的问题（假设n是偶数）。通过这种方式，计算的大小遵循 2 的连续幂的级数：追溯到最基本的初始问题的维度的增长每次都会翻倍。一方面，该算法让人想起古埃及赫利奥波利斯的 Ennead 系谱树的神学家，如开罗博物馆的 Petamon 石棺（编号 1160）所描述：
The phenomenon of growth is not a marginal issue, because it is involved in the most intimate structure of the algorithm. The modality of growth has a decisive role in determining efficiency. When it is explained that an algorithm functions in a recursive way, what is meant by this is more or less the following: the problem that the algorithm is designed to solve (for instance, the product of two numbers of sixteen digits each) is divided into problems of the same kind but of lesser dimension ( products of numbers with eight digits), and these are divided in turn into analogous problems that are smaller still ( products of numbers with four digits). The procedure of division continues until one reaches elementary problems of minimal dimension ( products of numbers with only one digit). From these latter, by way of an iterative calculus, one tracks back in the opposite direction until one solves the problem that was initially posed. This method of organizing recursive calculus, through a hierarchy of problems of the same kind but of decreasing dimension, is called divide et impera (‘divide and conquer’), from the old Latin motto. But the criterion of division is not, in itself, always and necessarily efficient. What matters is the law that regulates the decrease and successive growth of dimensions. Cramer’s method, which retraces the calculation of the determinant of a matrix of dimension n to calculation of the determinant of matrices of dimension n - 1, is completely inefficient. A great variety of algorithms use dichotomous divisions instead, that is to say, the initial problem of dimension n is divided into problems of dimension n/2 (assuming that n is even). In this way, the size of the calculation follows a progression of successive powers of 2: the growth of the dimension that goes back to the most elementary initial problem doubles each time. In one respect, the algorithm recalls the theologians of Heliopolis’ genealogical tree of the Ennead, in ancient Egypt, as described on the sarcophagus of Petamon (no. 1160) in the Cairo Museum:
- 我是转变成二的那一，
- 我是变身为四的二，
- 我是变身为八的四，
- 在这之后我是一。
赫利奥波利斯也通过 2 的幂进行划分，即 2 0 = 1, 2 1 = 2, 2 2 = 4, 2 3= 8，并且在根据需要重复多次之后，通过沿着相反方向的路径重新获得该路径，与递归过程完全相同。实际上，在减少一个小维的初始核之后，递归演算包括维数的逐步增加，从而能够解决问题，也就是说，计算分配的函数。该过程从一个小的初始核重新开始，并成为一个保持形式不变的增长。在埃及，不仅是神学公式隐喻地诉诸 2 的幂的渐进增长的概念。计算两个数字乘积的算术公式通常基于类似的增长标准。1
Division also took place in Heliopolis by means of the powers of 2, that is to say, 20 = 1, 21 = 2, 22 = 4, 23 = 8, and, after having repeated it as many times as required, the one was regained by following a path in the opposite direction, exactly as in the recursive procedure. In effect, after the reduction of an initial nucleus of small dimension, the recursive calculus consists of the progressive augmentation of the dimension which enables the solution of the problem, that is to say, the calculation of the assigned function. The procedure restarts from a small initial nucleus and becomes a growth that maintains the form unaltered. In Egypt it was not only the theological formula that had recourse, metaphorically, to the idea of progressive growth by the powers of 2. The arithmetical formulas for calculating the product of two numbers was regularly based on an analogous criterion of growth.1
数字的增长也可以以不同的方式发生，也就是说，不是通过 2 的连续幂，例如，根据著名的斐波那契数列，其中每个不同于 1 的数字都等于前面两个数字的总和。根据斐波那契准则或其他类似的序列划分问题，可能会获得计算优势，如在有理函数（两个多项式的商）的并行计算（同时执行多个操作）的情况下所证明的那样。计算的工作可以通过在空间中展开的图来描述，并且在图的展开的形状中可以识别计算复杂度。增长的标准可能因情况而异，但它在确定引导我们得出最终计算结果的最快路线方面仍然具有决定性意义。
The growth of numbers can also occur with different modalities, that is to say, not by successive powers of 2, for example, according to the celebrated Fibonacci series, in which every number different from 1 is equal to the sum of the two preceding numbers. Dividing a problem according to Fibonacci’s criterion, or other, analogous successions, one may gain computational advantages, as demonstrated in the case of the parallel calculation (where numerous operations are executed simultaneously) of a rational function (the quotient of two polynomials). The working of the calculation may be described by means of a graph that unfolds in space, and in the shape of the unfolding of the graph it is possible to identify the computational complexity. The criterion of growth may differ from one case to another, but it is nevertheless decisive in defining the fastest course leading us to the final result of the calculation.
迭代和追求不变性是计算的必要成分，黄金法则。即使在网络搜索引擎使用的计算中，我们也可以看到一个例子。网络上的每一页或文档——在行星尺度上巨大的信息网络——都被表示为一个比例巨大的图形中的节点i，其中此类节点的数量非常大，其矩阵L等价于比例可以与数十亿行和列相关联（即使我们正在处理稀疏矩阵，也就是说，有许多 0）。矩阵L表征图，在某种意义上，元素l ij如果节点i和节点j之间存在连接，则等于 1，如果不存在，则l ij等于0。如果我们必须衡量一个节点在图上的重要性，我们会考虑在该节点上及时收敛的需求数量。然后可以在迭代计算中表示该标准，固定向量x 0的初始假设估计的重要性节点并通过连接节点的重要性值的线性组合用新向量x重复更新它- 那些返回到有问题的节点以及那些节点可能导致的那些。节点的重要性，也就是网络上的文档，取决于连接的数量。然后，更新表示为自向量 x 的近似迭代演算，该自向量x对应于取决于L的矩阵A的最大特征值（A替代L，目的是使迭代计算收敛）。鉴于A有一个结构可以确定它的最大特征值等于 1，我们得到Ax = x，也就是说x是A的一个不动点，一个在方向或长度上保持不变的向量，如下线性算子A的应用。因此，不变性标准主导了对 Web 问题解决方案的迭代计算，其中包括为页面分配主要或次要重要性，以便详细说明对一般查询的响应。
Iteration and the pursuit of invariability are the necessary ingredients, the golden rule, of computation. We can see an example of this even in the calculations which are employed by search engines on the Web. Every page or document of the Web – of the immense network of information on a planetary scale – is represented as a node i of a graph of enormous proportions, in which the number of such nodes is extremely large and to which a matrix L of equivalent proportions can be associated, with billions of rows and columns (even if we are dealing with sparse matrices, that is to say, with many 0s). The matrix L characterizes the graph, in the sense that the element lij is equal to 1 if there is a connection between the node i and the node j, and lij is equal to 0 if not. If we have to measure the importance of a node on the graph, we take into account the number of demands that converge in time on that node. This criterion may then be expressed in an iterative calculation, fixing a vector x0 of initial hypothetical estimations of the importance of the nodes and updating it repeatedly with a new vector x by way of linear combinations of the importance values of the connected nodes – those that lead back to the node in question and those to which that node may lead. The importance of the node, that is to say, the document on the Web, depends upon the number of connections. The updating is then expressed in the iterative calculus, approximated, of the autovector x corresponding to the maximum eigenvalue of a matrix A that depends on L (A substitutes for L with the aim of making the iterative calculation convergent). Given that A has a structure that allows it to be established that its maximum eigenvalue is equal to 1, we get Ax = x, which is to say that x is a fixed point of A, a vector that remains unchanged, either in direction or length, following the application of the linear operator A. Therefore, a criterion of invariability presides over the iterative calculation of the solution of the problem of the Web, which consists in assigning major or minor importance to a page in order to elaborate a response to a generic inquiry.
计算中涉及的矩阵理论主要归功于奥斯卡·佩隆 (Oskar Perron) 的天才，他在上世纪初对其进行了阐述，其原因与今天因其在经济中的大量应用而闻名的原因完全不同，人口的增长和核裂变模型。具有讽刺意味的是，尽管 Perron 完全不知道其未来的应用，但在 1972 年给他的一个学生的一封信中，他表达了他对信息技术发展的所有保留意见，这对历史具有讽刺意味。2他问自己，后者是否不是“间谍活动 [ Spionatik ]”的同义词或次要分支，显然暗指图灵致力于解密德国通信自己在第二次世界大战期间。另一个例子是数学对自然科学和人工科学的研究具有不合理和非自愿的功效，并且作为一种意想不到的工具来对抗无限增长的数字的混合。
The theory of matrices involved in the calculations is principally due to the genius of Oskar Perron, who elaborated it at the beginning of the last century, for reasons quite different from those for which it is famous today on account of its numerous applications to the economy, the growth of populations and models of nuclear fission. It is an irony of history that, though wholly unaware of its future applications, Perron in a letter to one of his students dated 1972 expressed all his reservations about the growth of information technology.2 He asked himself if the latter was not synonymous with or a secondary branch of ‘espionage [Spionatik ]’, with an obvious allusion to the decryption of German communications that Turing had devoted himself to during the Second World War. Another example of the unreasonable and involuntary efficacy of mathematics for the study of the natural and artificial sciences, and as an unexpected tool to counter the hýbris of numbers that grow immeasurably.
关于作者
About the Author
PAOLO Z ELLINI是罗马大学的数学教授，他的研究重点是数值分析和数学思想的演变。他是国际畅销书《无限简史》的作者。保罗住在罗马。
PAOLO Z ELLINI is a professor of mathematics at the University of Rome, where his research focuses on numerical analysis and the evolution of mathematical thought. He is the author of the international bestseller A Brief History of Infinity. Paolo lives in Rome.
也由保罗泽里尼
ALSO BY PAOLO ZELLINI
无限简史
A Brief History of Infinity
笔记
Notes
介绍
Introduction
- 1 . Eugene P. Wigner 等人似乎在他的著名文章“自然科学中数学的不合理有效性”中沿着这些思路思考，发表于1960 年第 13 期的纯粹数学和应用数学通讯。
- 2 . 正是图灵将算法作为一个过程的概念（早在 1937 年提出）扩展到数值算法和通过数字计算器执行运算。参见 A. M. Turing，“矩阵过程中的舍入误差”（1948 年），在A. M. Turing编辑的文集中。J. L. Britton（阿姆斯特丹：北荷兰，1992 年）。
一、抽象、存在与现实
1. Abstraction, Existence and Reality
- 1 . 参见 Arthur Schopenhauer, Preisschrift über die Freiheit des Willens (1839), in Die beiden Grundprobleme der Ethik , in Zürcher Ausgabe。Werke in zehn Bänden（苏黎世：第欧根尼，1977 年），第一卷。六，页。96：“每一个存在都以一个本质 为前提：也就是说，每一个现存的事物也必须是某物，才能具有确定的本质。后者不可能存在，但也什么都不是，也就是说，像Ens metaphysicum这样的东西。, 等同于除了存在之外什么都不是的事物, 剥夺了任何规定或品质, 因此无法以由此而来的确定方式行动: 就像没有存在的本质就不可能是真实的一样,没有存在就不能假设存在本质。（康德用 100 泰勒的著名例子证明了这一点）。
- 2 . 复数集合中包含的集合K是一个数域，如果K至少包含一个非空元素，并且通过四种有理运算（加法、减法、乘法和除法）对K的元素进行运算，我们仍然得到一个K元素。最终受到威胁的是与四种理性操作相关的封闭领域。
- 3 . 两个域是同构的，因此无法区分，如果从一个到另一个的双单义对应g将来自第一个域A的两个元素x和y之间的运算结果发送到类似运算的结果来自第二个域B的相应g ( x ) 和g ( y ) 。如果运算是求和，我们得到g ( x + y ) = g ( x ) + g ( y ）。' n维向量空间 [ sui reali ]' 的概念足以识别没有同构的单个数学对象，因为一个定理确定任何两个n维向量空间 [ sui reali ] 都是同构的。一个类似的情况与有理数Q 的领域有关，具有加法和乘法的普通运算，以及分数之间的顺序关系（如果nq < mp ，则n/m < p/q）。如果一个数字字段K具有加法和乘法运算以及有序关系的所有其他字段都包含在具有加法和乘法以及有序关系的所有其他字段中，则K与Q重合。参见 B. Artmann, Der Zahlbegriff (Göttingen: Vandenhoeck & Ruprecht, 1983), pp. 18-19。
- 4 . S. Körner，数学哲学（1960 年）（多佛：纽约，1986 年），p。36.
- 5 . W. V. Quine，从逻辑的角度来看（马萨诸塞州剑桥，哈佛大学出版社，1953 年）。另见 J. J. Katz，现实理性主义（马萨诸塞州剑桥：麻省理工学院出版社，1998 年），第 117-75 页。
- 6 . N. Goodman 和 W. V. Quine，“迈向建设性唯名论的步骤”，《符号逻辑杂志》，1947 年第十二期，p。105.
- 7 . 对所有其他人有效的是 Rafael Bombelli 在研究三次方程之后引入的复数示例，这是建立代数方程始终有解的基本代数定理的必要步骤。更准确地说：如果p ( x ) 是n > 0次多项式，其系数在域K中，则在K中存在一个元素z使得p ( z ) = 0。
- 8 . 这一现实主义定义归功于迈克尔·达米特，与 H. Putnam，Mathematics, Matter and Method（剑桥：剑桥大学出版社，1975 年），第 69-70 页有关。
- 9 . 论文由 Putnam 提出，同上，第 70 页。
- 10 . 同上，p. 70. 对于迈克尔·达米特来说，普特南提醒我们，在哲学现实主义的表述中，一个基本要素涉及对我们思想之外的外部世界的感知。
- 11 . Simone Weil，手册，在Oeuvres完成，卷。VI，第三部分，编辑。F. de Lussy（巴黎：Gallimard，2002 年），p。179.
- 12 . Bertrand Russell，《数学哲学导论》（伦敦：Allen & Unwin，1919 年）。
2. 众神的数学
2. Mathematics of the Gods
- 1 . G. Scholem, Über einige Grundbegriffe des Judentums（法兰克福：Suhrkamp，1970 年），第 90-91 页。
- 2 . Dedekind 用他的递归理论在他的作品Was sind und is sollen die Zahlen? （布伦瑞克：Vieweg，1888 年）。
- 3 . 参见 A. Bürk, 'Das Āpastamba-Śulba-Sūtra', 在Zeitschrift der Deutschen Morgenländischen Gesellschaft , LVI, 1902, p. 336. 同样重要的是 A. Seidenberg 的文章，从“几何的仪式起源”开始，在“精确历史档案”中 《科学》，I，1961 年，我们还发现引用了 G. Thibault 对《博大传》的研究，“《博大传》，附有Dvárakánáthayajvan的注释”，载于Pandit，IX，1874；十，1875；NS，我，1876-77。
- 4 . 有关晷针的定义，请参阅下面的第 5 章，其中将阐明吠陀祭坛的几何形状与古代和现代数值算法之间的关系。
- 5 . G. Thibaut，“论 Śulvasútras”，孟加拉亚洲学会杂志，XLIV，1875 年，p。242.
- 6 . 这是数学史上重要的一段。有关更详细的方法，请参见 D. T. Whiteside，“17 世纪后期数学思想的模式”，《精确科学史档案》，I，1961 年，第 205-7 页。
- 7 . 参见例如 G. W. Leibniz, 'Historia et origo calculi differentis' (1714-16), 在Mathematische Schriften , vol. 五，编辑。C. I. Gerhardt (Hildesheim: Olms, 1962)。
- 8 . C. B. Boyer，《微积分的历史及其概念发展》（纽约：多佛，1959 年），p。118.
- 9 . “自我存在使外部访问点 [ parāñci khāni ] 无法 [捕获它]：结果 [个体存在] 只看到外部事物，而不是内部 ātman”，Śankara, Katha Upanishad , II, 1, 1 . 关于外向的欲望 ( kāma )，表达外部条件，而不是指向ātman的欲望，请参阅Taittirīya Upanishad，附有 Śankara 的评论（罗马：Āśram Vidyā，2006 年），第 42、99 和 114 页。在此很难不想起 Brouwer，这位伟大的荷兰数学家负责 19 世纪一些最重要的数学发现. Brouwer 是一种新数学的创始人，该数学被认为是一种内省行为的成果——然而，它并没有危及其公式的客观性。
- 10 . 有关详细信息，请参阅下面的第 4 章。
- 11 . Boethius，德机构算术，编辑。G. Friedlein（莱比锡：Teubner，1867 年），p。12：'... eum quoque numerum necesse est in propria semper sese habentem aequliter substantia permanere, eumque compositum non ex diversis ... sed ex ipso videtur esse compositus '。
3. 数学和哲学公式
3. Mathematical and Philosophical Formulas
- 1 . I. Thot，I paradossi di Zenone nel 'Parmenide' di Platone（那不勒斯：Bibliopolis，2006 年），p。18.
- 2 . K. Gaiser, Platons ungeschriebene Lehre (斯图加特: Klett, 1963), p. 15.
- 3 . 对于 Croce 对这个表达的使用，特别是关于他与 Enriques 的论战，参见 L. Russo 和 E. Santoni，Ingegni minuti: una storia della scienza in Italia（米兰：Feltrinelli，2010）。
- 4 . O. Neugebauer，古代精确科学（纽约：多佛，1969 年），p。35.
- 5 . A. P. Youschkevitch, Les Mathématiques arabes (VIII–XV siècles) , (Paris: Vrin, 1976), p. 47.
- 6 . 我们推测的引导划线的方法可以概括为迭代公式x k+ 1 =
4. 增长与减少，数量与性质
4. Growth and Decrease, Number and Nature
- 1 . Simone Weil 认为数学的每个方面都可能与这两个主要点有关，Cahiers，在Oeuvres complètes , vol. VI，第一部分，编辑。F. de Lussy（巴黎：Gallimard，1944 年），p。129.
- 2 . 根据 E. Bréhier 的翻译，引用于亚里士多德，La Métaphysique编辑。J.经编（巴黎：Vrin，1970 年），第一卷。我，页。23.
- 3 . Chantraine 将ousía翻译为“现实”、“实质”、“本质”。
- 4 . 马丁海德格尔，'Vom Wesen und Begriff der Phýsis。Aristoteles, Physik B, 1' (1939), in Gesamtausgabe (Frankfurt: Klostermann), vol. IX：Wegmarken，编辑。F.-W。冯赫尔曼，1976，p。272.
- 5 . 同上，第 278-9 页。
- 6 . K. Kerényi, Dionysos (1976)（斯图加特：Klett-Cotta，1994），p。25.
- 7 . 努梅纽斯，神父。2个地方。
- 8 . 阿基米德，De sphaera et cylindro，II，p。65，RR。15–16 穆勒。
- 9 . C. B. Boyer, The History of the Calculus and Its Conceptual Development (New York: Dover, 1959), pp. 193-4。
- 10 . 对于此类定义，可以查阅剑桥大学图书馆中未发表的手稿。特别是，例如，牛顿论文，女士。添加 3963, f。47r。有关更广泛的讨论，请参见 D. T. Whiteside，“17 世纪后期数学思想的模式”，《精确科学史档案》，I，1961 年，p。375.
- 11 . 在柏拉图的思想中，灵魂具有维持和凝聚身体的基本原则，否则这些身体会不断变化并最终分解。见 Numenius，fr。4b Des Places。
5. Katà gnómonos phýsin：Gnomon 的本质
5. Katà gnómonos phýsin: The Nature of the Gnomon
- 1 . A. Seidenberg，“几何的仪式起源”，精确科学史档案，I，1961 年，p。509.
- 2 . Marcel Proust，Du côté de chez Swann，在À la recherche du temps perdu编辑。J.-Y。Tadié（巴黎：Gallimard，第一卷，1987 年），p。182.
- 3 . 我们正在处理所谓的牛顿方法。为了计算函数f ( x ) 的局部最小值，其中x是n 个实分量的向量，我们在函数减小的方向d上逐步增加x，因为可以从通过小动作爬上一座小山，每个人都选择一个下坡方向。迭代总结在公式中：x k+ 1 = x k + λ k d k ，分配初始值x 0，其中λ k 是一个数字，它为每k个固定方向d k上的通道长度。它表明如果d k是f的任何下坡方向，使得f在x k+ 1中假定的值低于在x k中假定的值，则d k是线性方程组的解，即系数A k是对称的和正的。然后该方法采用以下形式：x k + 1 = x k +A k -1 g k，其中A k -1是 A k 的倒数，g k是在x k中计算的f的素数导数的向量（所谓的梯度）。该公式概括了求解方程f ( x ) = 0 或者更确切地说（对于n = 1）
的牛顿模式
xk+1=xk−f(xk)f′(xk), that derives in turn from the technique of gnomonic growth of the square. See C. Di Fiore, S. Fanelli and P. Zellini, ‘Low Complexity Secant Quasi-Newton Minimization Algorithms for Nonconvex Functions’, in Journal of Computational and Applied Mathematics, CCX, 2007, p. 172.
- 4 . 最简单的例子是数字m的平方根。如果m = 2，牛顿法包括计算近似的分数
6. Dýnamis：生产能力
6. Dýnamis: The Capacity to Produce
- 1 . T. L. Heath，欧几里得的“元素”十三卷书评注（纽约：多佛，1956 年），第一卷。我，页。348.
- 2 . 数学楔形文字，编辑。O. Neugebauer、A. Sachs 和 A. Goetze（纽黑文：美国东方学会和美国东方研究学院，1945 年），p。130.
- 3 . J. Høyrup，“毕达哥拉斯‘规则’和‘定理’——巴比伦数学与希腊数学之间关系的镜子”（罗斯基勒：罗斯基勒大学，1999 年），www.Academia.edu/3131799/。
- 4 . 参见 A. Bürk, 'Das Āpastamba-Śulba-Sūtra', 在Zeitschrift der Deutschen Morgenländischen Gesellschaft , LV, 1901, p. 556 和 LVI，1902，p。329; O. Becker, Das mathematische Denken der Antike (1957) (哥廷根: Vandenhoeck & Ruprecht, 1966), p. 33.
- 5 . Bürk，“Das Āpastamba-Śulba-Sūtra”，LVI，1902 年，p。327.
- 6 . J. Høyrup, ' Dýnamis , the Babylonians, and Theaetetus 147c7–148d7, in Historia Mathematica , XVII, 1990, p. 208. Høyrup 指出 Taisbak 将 dýnamis 解释为“扩展”。
- 7 . Bürk，“Das Āpastamba-Śulba-Sūtra”，LVI，1902 年，p。329.
- 8 . Martin Heidegger，“Die Frage nach der Technik”（1953 年），载于Gesamtausgabe（法兰克福：Klostermann），卷。VII: Vorträge und Aufsätze编辑。F.-W。冯赫尔曼，2000，第 11 页。
- 9 . Iamblichus，在 Nicomachi 算术介绍中，11 Teubner。
- 10 . D. J. O'Meara，毕达哥拉斯复兴（牛津：克拉伦登出版社，1989 年），p。44.
- 11 . Simone Weil，手册，在Oeuvres完成，卷。VI，第二部分，编辑。F. de Lussy（巴黎：Gallimard，1997 年），p。74.
- 12 . 计算由对角线和横向数定义的关系来近似
- 13 . O'Meara，毕达哥拉斯复兴，p。62.
- 14 . M. Psellus, 'Physical Numbers', 论毕达哥拉斯主义 V-VII，同上，第 218-19 页。
7.中场休息：精神力学
7. Intermission: Spiritual Mechanics
- 1 . B. Snell，Die Entdeckung des Geistes（汉堡：Claassen & Goverts，1946 年）。
- 2 . Simone Weil，手册，在Oeuvres完成，卷。VI，第四部分，编辑。F. de Lussy（巴黎：Gallimard，2006 年），p。336.
- 3 . J·W·冯·歌德，《Gott und Welt》，Urworte。Orphisch'，在Werke。汉堡包 Ausgabe 在 14 Bänden编辑。E. Trunz（慕尼黑：dtv，1998 年），卷。我，页。359.
- 4 . 弗里德里希·尼采 (Friedrich Nietzsche)，Al di là del bene e del male ( Beyond Good and Evil )，在Opere编辑。G. Colli 和 M. Montinari，第一卷。VI, pt II (米兰: Adelphi, 1968), p. 13.
8.芝诺悖论：运动的解释
8. Zeno’s Paradoxes: The Explanation of Movement
- 1 . T. L. Heath，希腊数学史，第一卷（纽约：多佛，1981 年），p。279; H. Weyl，数学哲学和自然科学（普林斯顿：普林斯顿大学出版社，1949 年），p。42; A. Grünbaum，“可以在有限时间内执行无限的操作吗？”，在《空间和时间的哲学问题》中（Dordrecht：Reidel，1973 年）。
- 2 . 线段除法是有限的或无限的 ( katà diaíresin )，而直线的末端是无限的 ( toîs eschátois )。
- 3 . 希思，希腊数学史，卷。我，页。276.
- 4 . 地点。同上。
- 5 . A. N. Whitehead，《过程与现实》（1929 年）（纽约：自由出版社，1969 年），第 84-5 页。
- 6 . F. Cajori，“芝诺运动论证的历史”，美国数学月刊，XXII，1915 年。
- 7 . Bertrand Russell，数学原理（剑桥：剑桥大学出版社，1903 年）。
- 8 . G. Colli，Zenone di Elea（米兰：Adelphi，1998 年），p。121.
- 9 . 引自 Whitehead, Process and Reality , p。84.
- 10 . 同上，p. 76.
- 11 . Henri Poincaré, La Science et l'hypothèse (1902) (Paris: Flammarion, 1968), p。51.
- 12 . 号码
- 13 . 怀特黑德，过程与现实，p。89.
- 14 . D. Hilbert 和 P. Bernays，Grundlagen der Mathematik，第一卷。1（柏林：斯普林格，1934 年），第 15-17 页，引自 S. C. Kleene，元数学导论（纽约：Van Nostrand，1952 年），第 54-5 页。
9. 多元化的悖论
9. The Paradoxes of Plurality
- 1 . Euclid, Elements , I, 1：“点是没有部分的点。” 欧几里得使用的术语semeîon似乎赋予该点一定程度的实在性，该程度低于亚里士多德的stigmé，后者指的是一种签名 [ puntamento ]。参见 T. L. Heath，欧几里德的“元素”十三本书（纽约：多佛，1956 年），第一卷。我，第 155-6 页。
10. 有限与无限：不可通约性与算法
10. The Limited and the Limitless: Incommensurability and Algorithms
- 1 . J. Stenzel, Zahl und Gestalt bei Platon und Aristoteles (Leipzig–Berlin: Teubner, 1924)。
- 2 . Mānava-Śrautasūtra，反式。J. M. van Gelder，Śata-Piṭaka 系列，卷。XXVII（新德里：国际印度文化学院，1963 年），第 10 页。308.
- 3 . 同上，p. 300. 这个公式让人想起在对话中反复出现的类似柏拉图式的“或多或少”公式。
- 4 . W. Knorr，“亚里士多德和不可通约性：一些进一步的思考”，精确科学史档案，XXIV，1981。
- 5 . 一个人得到l = d ´ + l ´ 因为d ´，Q 的对角线 DF，等于对角线l ´的平方的边的两倍
- 6 . 一方面，与穷举法类似的荒谬推理将依赖于阿基米德的假设（“给定两个齐次量级，总是存在一个较小的大于较大的倍数”），欧几里得定理由此推导出（元素，X，1）：“如果从较大的减去大于一半的量，则设置两个不相等的量，并且从剩下的量级大于它的一半，那么剩下的量级就会小于设定的较小量级。在正方形的情况下，如果存在共享度量m，则使用antanaíresis将计算大于 0 且小于m且可被m 整除的余数: 荒谬。阿基米德的命题排除了无限小数的存在，这些无限小数加上任意次数，永远不会超过指定的线。这些实体中没有一个可以定义两个量级的共同度量。
- 7 . 我们精确地有：d = l + l ´ = 2 l ´ + d ´，l = l ´ + d ´，和l ´ = d - l，d ´ = 2 l - d，其中d，l，d ´ 和l ´ 分别是较大和较小正方形的对角线和边。分量d和l的向量由矩阵M相乘得到元素 1 和 2 在第一行，元素 1 和 1 在第二行作为分量d ' 和 l ' 的向量，而分量 d ' 和l '的向量 是通过将M的倒数乘以分量d和l的向量。对角线和边之间的几何关系表明了一个数字级数。现在可以递归地计算数字 d和l，为每个数字固定一个等于 1 的初始值，并在渐进式增长的意义上使用前面的关系的正方形。第一个计算值将分别是 3 和 2。关系d : l然后假定值 1:1、3:2、7:5、17:12、41:39，其中数字d和l 增加根据上述递推规律。
- 8 . 可以认为是关系 d:l也可以解释为分数 d / l。这里隐含使用等于 1/ l的测量单位。分数 17/12 表示等于 17 乘以等于 1/12 的测量单位的量。作为l的增长表现出来，测量单位 1/ l减小， d / l和之间的距离也是如此
- 9 . 参见我的Gnomon（米兰：Adelphi，1999 年），第 343-4 和 384-5 页。
- 10 . 西蒙娜·威尔（Simone Weil）以精确的神学支持很好地强调了这一点：“圣约翰并没有断言：我们会因为看到上帝而快乐，而是：我们将与上帝相似，因为我们会看到他的本来面目（约翰一书，三，2）。我们将是纯粹的好人。我们将不复存在。但在这善尽头的虚无中，我们将比我们尘世生活中的任何时刻都更加真实。而处于邪恶边缘的虚无则没有现实。现实和存在是两件不同的事情。（ 《手册》，在Oeuvres complètes ，第 VI 卷，第 IV 部分，F. de Lussy 编辑（巴黎：Gallimard，2006 年），第 214 页）。
- 11 . 得到d 2 = 2 l 2 ± 1 ，因此关系d 2 : l 2 在l的无限增长时与2收敛。
- 12 . Simone Weil，手册，在Oeuvres完成，卷。VI，第三部分，编辑？（巴黎：Gallimard，2002 年），p。139.
- 13 . I. Thomas，希腊数学著作（1939 年）（马萨诸塞州剑桥：哈佛大学出版社，1967 年），第一卷。我，第 134-5 页。
- 14 . ( d 1 , d 2 ... d t ) B r形式的机器编号，其中B是数字表示的基础。在这种情况下，数字的大小不是由数字的数量表示，而是由指数r表示。
- 15 . 计算过程的稳定性取决于它在单个算术运算上传播由于舍入引起的误差的方式。这种操作的数量在很大程度上是人类受试者无法访问的，通常是非常高的数量级。
- 16 . 例如，如果a和b是非常大的数字，在第一个过程中，乘积 ( a × b ) 对于计算器的内存来说可能是一个太大的数字，因此会导致溢出。_ 在c是足够小的数字的情况下，可以在第二个过程中避免这种会停止该过程的不便。
- 17 . A. Bürk，“Das Āpastamba-Śulba-Sūtra”，载于Zeitschrift der Deutschen Morgenländischen Gesellschaft，LV，1901 年，p。557. 奥古斯丁在独白中的思想有着惊人的相似之处，他解释了如何从几何图形的存在中推断出真理的存在和灵魂的不朽，或者实际上是概念化它们的智慧。柏拉图的有关记忆的论述鼓励了类似的结论，该论述在将正方形加倍的过程中说明（Meno，85 d）。
11. 数的实在：康托的基本数列
11. The Reality of Numbers: Cantor’s Fundamental Sequences
- 1 . Georg Cantor, 'Extension d'un théorème de la théorie des séries trigonométriques', in Acta Mathematica , II, 1883, p. 337.
- 2 . B. Artmann, Der Zahlbegriff（哥廷根：Vandenhoeck & Ruprecht，1983 年），“Abschlussbemerkungen zu Kapitel 2”。
- 3 . 地点。同上。
- 4 . Georg Cantor，“Fondements d'une théorie générale des ensembles”，载于《数学学报》，II，1883 年，p。390.斜体是我的。
- 5 . P. E. B. Jourdain，Georg Cantor 的“介绍”，对建立超限数理论的贡献（纽约：Dover，1955 年），p。67.斜体是我的。
- 6 . J. W. Dauben，Georg Cantor：他的无限数学和哲学（马萨诸塞州剑桥：哈佛大学出版社，1979 年），第 126-8 页。
- 7 . Gottlob Frege, Die Grundlagen der Arithmetik (Breslau: Koebner, 1884)。
- 8 . 伊曼纽尔·康德，《批判性批判》 。Zweite Auflage 1787 年，在Werke。Akademeausgabe，卷。III（柏林：Reimer，1904 年），p。301。
- 9 . Henri Bergson, La Pensée et le mouvant , in Oeuvres , ed。A. Robinet（巴黎：法兰西大学出版社，1959 年），p。1254.
- 10 . 同上，p. 1257.
- 11 . Henri Bergson, Essai sur les données immediates de la conscience , in Oeuvres , p. 72.
- 12 . N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series (1942: 第一个发行量有限的版本；和 1949: 更扩展的版本，包括 N. Levinson 于 1943 年以期刊形式出现的文章) (Cambridge, Mass. ：麻省理工学院出版社，1949 年）。托普利茨矩阵在 Levinson 的一篇文章中进行了专题介绍，该文章简化并阐明了 Wiener 原始工作的许多方面：“滤波器设计和预测中的 Wiener RMS（均方根）误差标准”，《数学与物理学杂志》，XXV，1946 年，重印作为 Wiener，平稳时间序列的外推、插值和平滑中的附录B。
- 13 . n行n列的矩阵A的逆矩阵B使得B乘以A等于单位矩阵I，其主对角线上为 1，其他位置为 0。因此我们写B = A -1。如果A是线性方程组的系数矩阵，则了解其逆A -1使我们能够轻松地将系统的解计算为A -1对已知项向量的简单乘积。见下文第 16 章注释 1。
12. 数字的真实性：戴德金的部分
12. The Reality of Numbers: Dedekind’s Sections
- 1 . 参见 R. Smith，“亚里士多德三段论的数学起源”，《精确科学史档案》，1978 年第十九期。
- 2 . H. Reichenbach，Philosophie der Raum-Zeit-Lehre（柏林-莱比锡：de Gruyter，1928 年），par。9. Reichenbach 认为欧几里得数学的规范价值在于逻辑演示的意义远远超过图形的可视化构造。可以通过类比的方式来思考欧几里得元素的数学是几何代数的讨论。
- 3 . 参见例如 R. Courant 和 H. Robbins，什么是数学？（伦敦：牛津大学出版社，1941 年）。
- 4 . Richard Dedekind 是第一个构想实数理论的人，他指出（在 1858 年）具有决定性的完整性属性。他的基础论文《连续性和无理数》（1872 年）简要总结了他的基本思想。
- 5 . 参见 T. L. Heath，《欧几里德十三卷书评注》（纽约：多佛，1956 年），第一卷。二，页。125.
- 6 . 说两个量值 a和b之间的关系a : b小于两个整数m和n之间的关系m : n意味着na < mb。这同样适用于“等于”和“大于”的关系。
- 7 . 有关前欧几里得数学中的antanaíresis的全面讨论，请特别参见 D. H. Fowler，柏拉图学院的数学（牛津：克拉伦登出版社，1990 年）。
- 8 . B. Riemann, Über die Hypothesen, welche der Geometrie zu Grunde liegen (Göttingen: Dieterichsche Buchhandlung, 1867)。
- 9 . 见上文，注 6。
- 10 . 参见 G. H. Hardy 和 E. M. Wright，数论导论（1938 年），第五版（牛津：牛津大学出版社，1979 年），第 138-9 页。
- 11 . 西蒙娜·威尔，作品完成，卷。七，第一部分，编辑。F. de Lussy（巴黎：Gallimard，2012 年），p。465.斜体是我的。关于这个主题，请参阅 Roberto Calasso，Il Cacciatore Celeste（米兰：Adelphi，2016 年），第 272.
- 12 . A. M. Turing，“论可计算数，以及对Entscheidungsproblem 的应用”，在Proceedings of伦敦数学会，第二辑，XLII，1937；'A Correction'，同上，第二系列，XLIII，1938 年。
- 13 . W. V. Quine，集合论及其逻辑（马萨诸塞州剑桥：哈佛大学出版社，1963 年）。
- 14 . 另见 S. C. Kleene，数学逻辑（纽约：威利，1967 年），Ch. 2.
- 15 . K. Scheel, Der Briefwechsel Richard Dedekind–Heinrich Weber编辑。T. Sonar 和 K. Reich（柏林：de Gruyter，2014 年），p。277. 添加的重点是我的；B. Artmann, Der Zahlbegriff (哥廷根: Vandenhoeck & Ruprecht, 1983), p. 65.
- 16 . 在代数集（代数方程的根，即等于 0 的多项式的根）和形式为 a × 10 - k的十进制数系统之间存在同构，其中a为整数，k自然数之间的变量，代数数集也有一些漏洞：同上，p。42.
- 17 . Georg Cantor, 'Extension d'un théorème de la théorie des séries trigonométriques', in Acta Mathematica , II, 1883, p. 340。
- 18 . Georg Cantor，“Fondements d'une théorie générale des ensembles”，同上，p. 393.
- 19 . 同上，p. 403.
- 20 . 希腊数学家已经确定了阿基米德性质，当给定x和y时，当x小于y时， x的倍数优于y。证明的是一个有序且完整的数值域满足这个性质。
- 21 . Artmann, Der Zahlbegriff , p. 33.
- 22 . U. Dini，Fondamenti per la teorica delle funzioni di variabili reali（比萨：Nistri，1878 年），p。4.
- 23 . 同上，p. 6.
- 24 . A. N. Whitehead 和 Bertrand Russell，《数学原理》 （剑桥：剑桥大学出版社，1973 年），第 71-2 页。
- 25 . 奎因，集合论及其逻辑，p。3.
- 26 . J. W. Dauben，Georg Cantor：他的无限数学和哲学（马萨诸塞州剑桥：哈佛大学出版社，1979 年），第 221-2 页。
- 27 . 弗里德里希·尼采（Friedrich Nietzsche），Frammenti postumi，1887-1888 年，在Opere编辑。G. Colli 和 M. Montinari，第一卷。VIII, pt II (米兰: Adelphi, 1974), p. 47.
13. 数学：发现还是发明？
13. Mathematics: A Discovery or an Invention?
- 1 . 理查德·戴德金德，扎伦死了吗？（布伦瑞克：Vieweg，1888 年），p。iii.
- 2 . 同上，p. 六。斜体字是我的。
- 3 . Richard Dedekind，Stetigkeit un irrationale Zahlen（布伦瑞克：Vieweg，1872 年），第 18 和 20-22 页。
- 4 . S. Pincherle，“Saggio di una introduzione alla teoria delle funzioni analitiche secondo i principii del Prof. C. Weierstrass”，载于Giornale de Matematiche，XVIII，1880 年，第 186、190 和 191 页。
- 5 . 引自 F. Cajori，数学符号史(1928-9)（纽约：多佛，1993 年），第一卷。二，页。333. 有人被提示与贝内代托·克罗齐对哲学家的劝告进行比较：“思考，不要计算！[ Qui incipit numerare, incipit errare！ ]'：Benedetto Croce，'Filosofia della pratica。Economia ed etica'，在Filosofia dello spirito中，卷。III（巴里：Laterza，1909 年，作者修订的第二版，1915 年），p。270. 这其中隐含了一种类似于怀特黑德的观察，但后来变成了否定的。
- 6 . G. H. Hardy，数学家的道歉（剑桥：剑桥大学出版社，1940 年）。
14. 从连续体到数字化
14. From the Continuum to the Digital
- 1 . Richard Dedekind，Stetigkeit und irrationale Zahlen，（布伦瑞克：Vieweg，1872 年），第 18-19 页。
- 2 . “Das Kontinuum als Medium freien Werdens”是 Hermann Weyl 的表述。参见 H. Weyl，“Über die neue Grundlagenkrise der Mathematik”，《Mathematische Zeitschrift 》 ，X，1921 年。
- 3 . 弗里德里希·尼采（Friedrich Nietzsche），Frammenti postumi，1888-1889年，在Opere编辑。G. Colli 和 M. Montinari，第一卷。VIII, pt II (米兰: Adelphi, 1974), p. 162.
- 4 . 参见 D. Hilbert，“Über das Unendliche”，在数学年鉴，XCV，1926 年。
- 5 . 埃。Borel，“Les “Paradoxes” de la théorie des ensembles”，载于Annales scientifiques de l'École Normale Supérieure，第三辑，XXV，1908 年。
- 6 . 例如，参见 V. A. Uspensky，“Kolmogorov and Mathematical Logic”，在Journal of Symbolic Logic中，LVII，1992，p。393.
- 7 . 参见 E. Zermelo, 'Neuer Beweis für die Möglichkeit einer Wohlordnung', in Mathematische Annalen , LXV, 1908。关于冯诺依曼的理论，参见 P. R. Halmos, Naive Set Theory (New York: Springer, 1974)。
- 8 . 例如，对角线数字d和横向数字l就是这种情况。可以这样计算k，使得 2 与关系d 2 : l 2之间的距离小于一个ε，如果l大于k则任意小，因为我们知道距离等于
- 9 . G. Peano，“Interpolazione nelle tavole numeriche”，在Atti della Reale Accademia delle Scienza di Torino，LIII，1917-18，p。693.
- 10 . 参见 R. Courant 和 D. Hilbert，数学物理方法，第一卷。II：R. Courant 的偏微分方程（纽约：Interscience，1962 年），p。229.
- 11 . 同上，p. 230.
- 12 . B.帕莱特。“数值分析进展”，SIAM 评论，XX，1978 年，第 448-9 页。
- 13 . A. A. Markov Jr，“论构造函数”，美国数学会翻译，第二系列，XXIX，1963 年，第 163-4 页。
- 14 . A. A. Markov Jr，“算法理论”，同上，第二系列，XV，1960，p。3.
- 15 . H. Rogers Jr，递归函数和有效可计算性理论（纽约：McGraw-Hill，1967 年），p。31.
15. 数字的增长
15. The Growth of Numbers
- 1 . 有关详细信息，请参阅 R. Courant 和 H. Robbins，什么是数学？（伦敦：牛津大学出版社，1941 年），第 176-80 页。
- 2 . 参见 M. S. Paterson，'Efficient Iterations for Algebraic Numbers'，R. E. Miller 和 J. Thatcher (eds.)，Complexity of Computer Computations (New York–London: Plenum Press, 1972)。在p阶迭代方法中，它在每一步k计算一个有理表达式g ，它提供代数方程的根z的近似值p/q （该阶测量它逼近z 的速度），分母q，对于一个相当放大的k，超过s = r 的值L d sk对于每个r < p，其中L和d是常数， d大于 1。
- 3 . 例如，由于类似于舍入的自动过程，分数 1300/1113 可以近似为更简单的分数 7/6。通过将分子和分母除以相同的因子来简化分数是过分的。此外，两个随机整数之间有素数的概率很高，等于
- 4 . D. Knuth，计算机编程的艺术，第一卷。II（马萨诸塞州雷丁：Addison-Wesley，1969 年），p。292.
- 5 . 考虑一个单一的代数方程，由x中的多项式定义为 0。如果多项式的次数大于 4，则该方程通常不允许解析类型的解。因此，必须使用计算有理数序列的数值程序来近似解，这些有理数序列将这些解逼近到所需的精度。对于微分或积分方程，可以找到类似的情况，其解不能用解析公式表示，因此有必要对其进行数值近似。为此，有必要求解具有高维系数矩阵的线性方程组。
- 6 . 参见 W. Gautschi, 'Computational Aspects of Three-term Recurrence Relations', in SIAM Review , IX, 1967。例如，假设形式为y k+ 1 = ay k = by k − 1的三项递归关系，这使得可以计算y k，对于k = 2, 3, ...，如果它们被分配了初始值y 0和y 1，通常以不稳定现象为特征，也就是说，误差的不受控制的增长。使用可比较的迭代策略，以良好的顺序求解简单的微分方程系统，在这种情况下，不确定性关于初始值可能会导致由这种不稳定的数值现象引起的振荡级数中的项呈指数增长。关于这个主题，请参见 G. Dahlquist，“33 年的数值不稳定，第一部分”，位数值数学，XXV，1985 年，以及 G. Dahlquist 和 Åke Björck，数值方法（1974 年）（纽约：Dover，2003 年），第 342 和 373 页。
- 7 . 引自 A. Knoebel、R. Laubenbacher、J. Lodder 和 D. Pengelley，Mathematical Masterpieces（纽约：Springer，2007 年），p。65.
16. 矩阵的增长
16. The Growth of Matrices
- 1 . 向量w具有位置i的元素作为乘积a ik × v k的总和，当k变化时，其中v k是向量v的位置k的元素。类似地， n行n列的两个方阵A和B的乘积是方阵C，其维数与A和B相同，其中元素c ij是总和，当k变化，所有产品a ik × b kj。如果两个矩阵A和B的乘积是单位矩阵I，则B是A的逆矩阵，并且A称为可逆矩阵，我们写为B = A -1。具有可逆系数的矩阵A的线性方程组Ax = b的向量解x等于向量b的A -1的乘积. 见上文第 11 章注释 13。
- 2 . V. Y. Pan，“并行矩阵计算的复杂性”，理论计算机科学，LIV，1987 年。
- 3 . G. Strang，“小波”，美国科学家，LXXXII，1994 年。
- 4 . 参见 J. H. Wilkinson, 'Some Comments from a Numerical Analyst' (1970 Turing Lecture), in the Association for Computing Machinery , XVIII, 1971。关于矩阵计算发展的初始阶段以及 Whittaker 和 Robinson 的著作，见 O. Taussky，“我如何成为矩阵理论的火炬手”，美国数学月刊，XCV，1988 年。有关 Cramer 方法效率低下的第一个迹象，请参见 G. E. Forsythe，“求解线性代数方程可能很有趣”，美国数学会公报，LIX，1953 年。
- 5 . D. Hilbert, 'Ein Beitrag zur Theorie des Legendre'schen Polynoms', in Acta Mathematica , XVIII, 1894。
- 6 . 误差表达式包含变量x的幂，当x介于 0 和 1 之间时，它接近于线性相关。因此，将误差降至最低的策略返回到由矩阵H标识的线性方程组，其行接近线性相关，结果H的行列式接近 0，即H病态性质的标志。实际上，矩阵H的条件索引的倒数是具有空行列式的矩阵集。因此，这个距离越小，调节指数越大。参见 J. W. Demmel，“关于条件数和到最近不适定问题的距离”，Numerische Mathematik，LI，1987。
- 7 . 矩阵在 G. E. Forsythe, 'Pitfalls in Computation, or Why a Math Book is not Enough', American Mathematical Monthly , LXXVII, 1970 中以其先前的显式形式进行了描述。另见 G. E. Forsythe 和 C. B. Moler, Computer Solution of Linear代数系统(Englewood Cliffs, NJ: Prentice-Hall, 1967)。
- 8 . 有关定义向量矩阵乘积的表达式，请参见上面的注释 1。六方程组的解是通过将六行六列的矩阵H -1与向量b相乘，从而获得向量解x，其第五个元素x 5由包含项4410000的表达式给出b 5，其中b 5是向量b的第五个元素。如果b 5有一个小的变化，一个正增量，比方说，等于10 -6，而不是4410000 b 5我们将有 4410000( b 5 + 10 -6 ); 也就是说，仅由于干扰10 -6 ， x 5受到等于4.41的变化。对于H的倒数的任何足够高的元素，自然可以验证误差的等效放大。
- 9 . 有关详细信息，请参见 N. J. Higham，数值算法的准确性和稳定性（费城：SIAM，1996 年），p。177.
- 10 . 如果对于每个非空实向量x ， x T Ax > 0 ，则实对称矩阵A（一个，即a ij = a ji ）是正定的。矩阵A是正定的当且仅当其特征值为（实数和）正数。
- 11 . 更准确地说，如果X是A的计算倒数，则AX和单位矩阵I之间的距离不会大于该值
- 12 . J. H. Wilkinson，“现代错误分析”，SIAM 评论，1971 年，第十三页，第 3 页。550. Wilkinson 还强调了对 Wallace Givens 提出的误差分析的宝贵贡献，他是 1954 年第一个设想对误差进行反向分析的人，包括在矩阵中反映计算过程中累积的误差：近似算法被认为是一种算法，它不是对矩阵A而是对一个矩阵A + Δ A进行精确求逆，即一个被误差 Δ A扰动的矩阵。
- 13 . A. M. Turing，“矩阵过程中的舍入误差”（1948 年），载于A. M. Turing编辑的全集。J. L. Britton（阿姆斯特丹：北荷兰，1992 年）。
- 14 . 通常，已知迭代方法的效率，尤其是解决方案的收敛速度，取决于条件指数µ。
- 15 . A的欧几里得范数，也称为谱范数，因为如果A是对称且正定的其谱半径 [ raggio spettrale ]，即其最大特征值，小于或等于 Frobenius 范数。
- 16 . 可能是，但不一定如此。错误本身是不可知的；我们只能确定它的界限。但是，如果这些限制与一个大的误差相容，那么关于计算方法的可靠性的每一个确定性都会被破坏。仍然有调节方法，矩阵中的干预技术来修改特征值的分布。关于这个主题有大量的文献。
- 17 . 见上文，第 2 章注释 11。
- 18 . A. A. Markov Jr，“算法理论”，美国数学学会翻译，第二系列，XXIX，1963 年，p。1.
- 19 . H. H. Goldstine 和 J. von Neumann，关于大型计算机的原理，载于 John von Neumann，全集，卷。五，编辑。A. H. Taub（牛津：佩加蒙出版社，1963 年），p。3.
17. 基本面的危机和复杂性的增长：现实与效率
17. The Crisis of Fundamentals and the Growth of Complexity: Reality and Efficiency
- 1 . 矩阵A的行列式是 A 的元素的多项式函数……
- 2 . J. Edmonds，“不同代表系统和线性代数”，在国家标准局研究杂志，LXXI B，1967 年首次指出了这种潜在的不便。另见 L. Lovász， “经典数学中某些概念的算法方面”，载于 J. W. de Bakker、M. Hazenwinkel 和 J. K. Lenstra（编），数学和计算机科学（阿姆斯特丹：北荷兰，1986 年）。
- 3 . 参照。J. Hartmanis，“Gödel、von Neumann 和 P = ?NP 问题”，结构复杂性专栏，1989 年，EATCS 公报，第一卷。38，第 101-7 页。Hartmanis 准确地提到了三个基本问题——不完整性、不可判定性和复杂性——从而解释最后一个如何继承基本问题，在数学中，什么可能或可能不承认一个解决方案。
- 4 . 即使是简单的二次多项式p ( x ) 的计算也并非没有挑战。如果我们将方程y = p ( x ) = 1 + ( x - 5555.5) 2的简单抛物线公式写成p ( x ) = 30863581.25 - 11111 x + x 2，计算p (5555) 和p ( 5554.5）在具有六个有效数字的机器算术中，我们分别获得数字 0 和 100。换句话说，对x的小扰动可能会反映在p ( x )。该示例取自 J. R. Rice, Numerical Methods, Software, and Analysis (New York: McGraw-Hill, 1983), p。60.
- 5 . A. L. Cauchy，Cours d'Analyse de l'École Royale Polytechnique。首演派对。分析 Algébrique（巴黎：Debure，1821 年），p。463.斜体是我的。
- 6 . 如果x属于区间 [ c , d ]，则h ( x ) 的绝对值小于或等于ε。如果在同一区间t ( x ) 为正，我们不能推断h ( x ) 也一样，因为不等式h ( x ) + e ( x ) > 0 与不等式h ( x ) < 0相容。如果x不属于 [ c , d 则不同]。在这种情况下，h ( x ) 的绝对值大于ε，如果t ( x ) 为正，则h ( x ) 也为正。事实上，如果h ( x ) < 0，它应该是h ( x ) < − ε的情况，因此h ( x ) + e ( x ) < 0 也是矛盾的。
18. Verum et Factum
18. Verum et Factum
- 1 . Simone Weil，手册，在Oeuvres完成，卷。VI，第三部分，编辑。F. de Lussy（巴黎：Gallimard，2002 年），p。90.
- 2 . E. P. Wigner，“数学在自然科学中的不合理有效性”，在纯粹和应用数学的交流中，第十三期，1960 年。
- 3 . A. Capelli，“Saggio sulla introduzione dei numeri irrazionali col metodo delle classi contigue”，载于Giornale di Matematiche，XXXV，1897 年，p。210.
- 4 . 威尔，手册，第一卷。第六部分，第二部分，p。352.
- 5 . Thomas Aquinas, Summa Theologiae , I, q. 4，一个。1: ' Oportet enim ante id quod est in potentia, esse aliquid actu, cum ens in potentia non reducatur in actu, nisi per aliquod ens in actu。'
19. 递归和不变性
19. Recursion and Invariability
指数
Index
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- 抽象对象 14
- 抽象 11-12 , 15 , 155 , 155-6
- 现实 117
- 埃斯库罗斯 18 , 24
- 阿基瓦，拉比 21
- 代数
- 代数自动性 31
- 代数闭包 177
- 代数方程 30
- 三度 22
- 代数数 159–60
- 算法程序 104-13
- 算法结构 8 , 190
- 算法 4-6 , 7-8 , 97 , 147 , 154
- 分析， 153-4的算术化
- 解析式 151–2
- anatanaíresis 129–30 , 162
- 古代算术 4
- 安泰 155
- 安塔奈瑞西斯 37 , 107 , 111
- 反律， 120
- 意大利面 27 , 29 , 62 , 113
- 近似微积分 105-6
- 近似值 98 , 99 , 157 , 165
- 阿基米德 40 , 90 , 91 , 157
- 阿基米德 公设 211–12n6
- 阿奇塔斯 67
- 亚里士多德 37 , 43 , 45 , 46–8 , 51 , 67 , 72 , 73 , 74 , 78–80 , 86 , 88 , 89–90 , 90 , 93 , 95–6 , 100 , 112 , 119 , 125 , 129
- 算术
- 算术化 153–4 , 178–9
- 算术分析 22
- 军备研究部 169
- 天文标志 24
- 原子命题 13
- 原子论 145-6
- 原子现实主义 117 , 119
- 原子论 91 , 94–5 , 112
- 奥古斯丁，圣 91、134、214n17
- 自动计算 100
- 自动学习 60
- 公理 12
- 可计算性 8
- 191的限制
- 计算 23–4
- 计算器 5 , 15 , 164
- 微积分 4 , 5 , 15 , 23 , 35 , 53 , 153 , 168-9 , 180-1
- 康托尔, 乔治 19 , 22 , 84 , 86 , 92–3 , 96 , 98 , 114–24 , 126 , 132 , 133 , 134 , 137 , 146 , 149–50 , 159–60 , 177
- 卡尔达诺，吉罗拉莫 22
- 基数 92
- 柯西，奥古斯丁-路易斯 81 , 105 , 106 , 187
- 卡瓦列里，博纳文图拉 31
- 更改 44–5 , 48–9 , 50–1 , 54 , 58–9
- 中国数学 23
- 克吕西普斯 73
- 教堂，A. 185
- 类，关闭 22
- 巧合 66
- 罗德岛巨像 50–1
- 完整性 127 , 132–3 , 192 , 193
- 复数 22 , 119 , 203n7
- 复杂性 183–6
- 可计算性 5
- 计算 23
- 计算
- 计算复杂度 5 , 183–6 , 198
- 计算成本 156
- 计算效率 5–6
- 计算数学 150 , 178 , 186 , 191
- 计算过程 8-9 , 153 , 187
- 计算模式 6
- 计算策略 8 , 179
- 条件指数 174–6 , 224n14
- 意识 53 , 54 , 68
- 建构主义哲学 8
- 连续性 192
- 连续性， 41法则
- 连续延伸，雾化 84
- 连续分数, 129 , 163
- 连续数学 4
- 连续体, 7 , 18 , 114
- Cramer 算法 169 , 172–3 , 177 , 183 , 196
- 克罗齐，贝内代托 39
- 曲线-直线关系 43
- 控制论 122
- 小数 151
- 减少 43 , 44 , 45 , 49–50
- 戴德金, 理查德 15 , 22 , 64 , 75 , 84 , 86 , 96 , 98 , 107 , 116–17 , 118 , 120–1 , 125–38 , 139–40 , 141 , 144 , 146 , 148–9 , 177 –8 , 194–5 , 216n4
- 扣除 13
- 德谟克利特 94 , 95
- 演示程序， 102-4
- 密度 128 , 133
- 推导 22
- 笛卡尔，勒内 7 , 138
- 描述，理论 136
- 确定性混沌 161
- 对角线论证， 116-17
- 对角线法， 19
- 透析 110
- 辩证法 37
- 二分法 196-7
- 二分法， 100-2
- 微积分 31 , 58
- 数字计算 160–1 , 162–3 , 168
- 微积分 194
- 数字计算 165-6
- 数字革命 153–4
- 尺寸 92 , 166
- 迪尼，乌利塞 134-5 , 137-8
- 发现 15–16
- 神圣的 27
- 196-7师
- 到四肢 77–80
- 动力系统，理论 161
- 动力学 39
- 动态62–3 , 65 , 90 , 181
- 有效实现 116
- 效率 5–6 , 178–9 , 183 , 186–91 , 193–4 , 196–7 , 224n14
- 埃及 196–7
- 埃利亚 24
- Eleatic 逻辑 114
- 体现 122–3
- 谜团 26
- 放大 28 , 28–9 , 50–1 , 61
- Ennead， 197年的家谱树
- 枚举 18–19
- Epicharmus 44-5 , 74-5
- 方程
- 等价 27–8 , 94 , 110
- 埃拉托 色尼 26–7 , 50–1
- 本质 11-12
- 欧几里得 15 , 18 , 28 , 40 , 61 , 62 , 90 , 100 , 112 , 125–6 , 127 , 130 , 131 , 162 , 211n1
- 欧几里得算法, 37 , 102–4 , 111
- 欧几里得范数 176 , 224–5n15
- 欧几里得定理 57-9 , 212n6
- Eudoxos/Euclid 关系理论 128-9 , 141
- 欧拉，莱昂哈德 58 , 163 , 164
- 阿斯卡隆的 尤托修斯 26–7
- 用尽，方法 40
- 存在 108
- 存在 11-12
- 存在量词， 18
- 伽利略 16 , 38–9 , 41
- 高斯，卡尔·弗里德里希 172-3
- 高斯算法 172-3 , 183
- 几何图形，增长 7
- 几何形式，放大 6-7 , 8 , 50-1 , 61
- 几何代数 112
- 几何点， 51-2
- 几何 4 , 6-7
- 吉文斯，华莱士 224n12
- 晷针, 30 , 55–60
- 指针增量 30
- 哥德尔，库尔特 184–6
- 神 6-7 , 21
- 歌德，约翰·沃尔夫冈 74-5
- 戈德斯汀，赫尔曼 173–4 , 176 , 180 , 191
- 古德曼，N. 14
- 高尔吉亚 25
- 希腊，古 6 , 8 , 17–18 , 26–7 , 41–2 , 61 , 98 , 157 , 181
- 增长 43–54 , 60 , 196–200
- 扬布里库斯 25 , 44 , 67 , 109
- 无知 3
- 不适定问题 152
- 不变性 35
- 不可通约性 4 , 130
- 不可通约的数量， 41-2之间的关系
- 不确定性， 190的云
- 指数 115
- 不可分割的 31
- 无限, 17–18 , 43 , 45 , 69 , 76
- 无穷小数 80 , 81
- 无穷大 17–18 , 108
- 信息技术 154 , 199
- 信息论 5
- 不确定区间 187
- 间隔， 106-7的不定序列
- 直觉主义 147-50
- 不变性 45 , 198–9
- 发明 15–16
- 无理数 7–8 , 19 , 36 , 82–3 , 96 , 119 , 126–7 , 128–30 , 144–5 , 210n12
- 近似值 99–113 , 158–60 , 162–3
- 作为区间 106-7的不定序列
- 同构 12 , 117–18 , 202n3 , 217n16
- 迭代 198–9
- 拉格朗日，约瑟夫-路易斯 58 , 151
- 学习 60
- 勒贝格，亨利 144-5
- 莱布尼茨，W. 80 , 120 , 153
- 留基普斯 94 , 95
- 相似度 129
- 限制 53 , 104–5 , 188
- 限制，达到 80
- 线性代数 168
- 线性方程组 165–6 , 168 , 169 , 170 , 174–5 , 180 , 183
- 线性问题 166
- 线性化 165–6
- 第92 , 114行
- 约瑟夫·刘维尔 100 , 158–60
- 洛克，约翰 117
- 逻辑 12–14 , 18 , 125–6 , 131 , 135–7 , 137–8
- 逻辑主义 12-13
- 标志 36 , 47 , 66 , 72–3 , 97
- 卢克莱 修 94–5
- 马赫，恩斯特 15
- 震级 75
- 马拉穆德，查尔斯 25
- 马可福音 74
- Markov, Andrej A., Jr 155 , 178
- 重要性 46
- 数学抽象 9
- 数学分析 42
- 数学概念，行为 3
- 数学实体 11–12 , 16 , 17 , 191
- 数学领域 12
- 数学直觉主义 8 , 147–50
- 数学建模 151-5
- 数学对象 12
- 数学问题，解决方案 192-4
- 数学现实 5
- 数学思维 3 , 17
- 数学 11-12 , 22
- 矩阵和矩阵计算 215n12 , 215n13
- 萨摩斯的 梅利苏斯 76–7
- 心理框架 17
- 美索不达米亚数学 6 , 23 , 39 , 41-2 , 61 , 61-2
- 变态 48-9
- 形而上学 13 , 41 , 43
- 模型和建模 3 , 151–5
- 摩西和拉比阿基瓦， 21年的寓言
- 运动， 78定律
- 运动 73–4
- 芝诺的悖论 76–88
- 可变性 45
- 神秘主义 7 , 13
- 国家物理实验室 169
- 自然 8 , 69 , 181–2
- 新柏拉图主义者 69
- 纽格鲍尔, 奥托 41 , 42
- 神经网络 60
- 牛顿，I. 25 , 30 , 52–3 , 59–60 , 70 , 80 , 153 , 158 , 205n6
- 牛顿法 207n3
- 库萨的尼古拉斯 36–7
- 格拉萨的尼科马科斯 70
- 尼采, F. 20 , 75 , 137-8 , 145-6
- 唯名论 13-14
- 不存在 108
- 没什么 90
- NP 完全问题 183-4 , 185-6
- 数字 11 , 216n4
- 实际情况 118
- 代数 159–60
- 二进制 160–1
- 完整性 192
- 复合物 22 , 119 , 203n7
- 连续性 192
- 枚举 19
- 12的本质
- 几何形式 23
- 和成长 67
- 增长 6 , 7 , 97 , 157–64 , 197–8
- 无理数 7–8 , 19 , 36 , 82–3 , 96 , 99–113 , 106–7 , 119 , 126–7 , 128–30 , 144–5 , 158–60 , 162–3 , 210n12
- 和逻辑 13 , 14
- 和自然 69–70
- 本体状态 18 , 120 , 177
- 起源 141–2
- 毕达哥拉斯 55–7 , 56
- 有理数 22 , 82–3 , 107 , 128 , 145 , 149 , 161 , 164 , 192–3
- 实数 15 , 42 , 83 , 88 , 114–16 , 117–18 , 119 , 123–4 , 127–8 , 143 , 149–50 , 192
- 和现实 67-8
- 9 , 14 , 17–18 , 19 , 70 , 97 , 114–24 , 125–38 , 191的现实
- 38的意义
- 4的理论
- 超越 159–60
- 超限 118–20 , 137
- 75的不可控增长
- 整个 7 , 22 , 131 , 177–8
- 数值计算 162-3
- 数字字段 12 , 177
- 数值优化 59-60
- 数值程序 7
- 数字级数 51 , 90
- 数值系列，极限理论 78-9
- 帕拉梅德斯 24–5
- 帕莱特，贝雷斯福德 153
- 巴门尼德 25 , 77 , 97
- 巴门尼德的论文 114
- 激情 73
- 皮亚诺 ，朱塞佩22、148、151 _ _
- 看法 53 , 54
- 佩伦，奥斯卡 199–200
- 透视，颠覆 105-6
- 菲洛劳斯 26 , 55 , 57
- 哲学表达 36
- 哲学公式 36-42
- 哲学 180–1
- 物理现实 151
- 物理学 16–17 , 117 , 152
- 物理 45–9 , 51 , 59 , 65–6 , 70 , 73 , 75 , 90
- Pincherle，萨尔瓦多 141
- 柏拉图 17–18 , 24–5 , 37 , 38 , 48–9 , 51–2 , 53–4 , 63 , 65–7 , 72 , 74 , 90 , 95 , 97–9 , 98–9 , 109 , 132 , 140 , 181
- 柏拉图式的好 49
- 柏拉图式的愿景， 35
- 普劳图斯 192
- 普罗提努斯 68–9 , 70 , 131
- 复数, 悖论 89–96 , 114
- 庞加莱，亨利 85 , 107 , 140 , 145 , 146 , 188
- 点 90–3 , 114 , 118 , 126 , 211n1
- 电源 63
- 般若 32-3 , 35
- 可预测性 161
- 预先建立的和谐 146
- 第26条、第43-4条、第 61条、第65条、第90条、第97条、第 107条、第109 条
- 生产 64–5
- 进展 35
- 无限期 94
- 普罗米修斯 24
- 比例 35 , 39 , 125
- 变形虫 49
- 普鲁斯特，马塞尔 59 , 122
- Psellus，迈克尔 70
- 毕达哥拉斯 25 , 44–5 , 45 , 67
- 毕达哥拉斯定理：28 , 61-3 , 99 , 113
- 毕达哥拉斯数字 55–7
- 毕达哥拉斯主义 4 , 41
- 毕达哥拉斯 4 , 23 , 26 , 37 , 38 , 95–6 , 119
- 约瑟夫·拉夫森 30 , 57 , 59 , 205n6
- 理性语料库， 107
- 有理对角线, 98 , 109–10
- 有理数 22 , 82–3 , 107 , 128 , 145 , 149 , 161 , 164 , 192–3
- 实函数， 134-5的理论
- 实数 15 , 42 , 83 , 88 , 114–16 , 117–18 , 119 , 123–4 , 127–8 , 143 , 149–50 , 192 , 216n4
- 现实主义 16-17 , 135-7
- 现实 3 , 4 , 11 , 15 , 17–18 , 90 , 94–5 , 98 , 108–9 , 181–2 , 192
- 可实现性 155–6 , 177
- 反复出现的关系 163-4
- 递归 196-7
- 148-9理论
- 递归演算 196-7
- 还原论 181
- 法尔西, 39-40
- 赖兴巴赫，H. 215–16n2
- 关系 17-18 , 39 , 47-8 , 68 , 99
- 关系 4 , 66-7
- 关系
- 130的平等
- 128-9的Eudoxos /Euclid理论
- 13的现实
- 104-6的序列
- 宗教和数学 24-6
- 宗教仪式 7
- 文艺复兴， 30
- 残差 106
- 理查悖论 147
- 黎曼，伯恩哈德 129
- 仪式 8
- 罗宾逊，G. 168-9
- 罗杰斯，哈特利 155-6
- 舍入 220–1n3
- 罗素，伯特兰 12-13 , 13-14 , 18 , 125 , 127 , 131 , 146-7
- 萨克斯，亚伯拉罕 41
- 桑卡拉 33
- Śatapatha 婆罗门 25 , 31-5
- 怀疑唯名论 135-6
- 图式 6-7 , 8
- 科学发现 15-16
- 科学理论 16
- 第 64条、第127条、第131-2条、第134条、第 194-5 条
- 语义悖论 5
- 集合论 22
- 形状 26 , 68–9
- 相似度 94
- 斯内尔，布鲁诺 73–4
- 苏格拉底 65 , 72
- 灵魂, 37 , 55 , 206n11
- 空间
- 时空 83
- 时空连续体， 83
- 速度 82
- 精子 36 , 69
- 斯宾诺莎，巴鲁克 73 , 181–2
- 正方形，放大 29–30
- 平方根 59 , 158 , 208n4
- 稳定性 58 , 213n15
- 斯多葛学派 37 , 69 , 73
- 直线数关系 43
- 斯特朗，吉尔伯特 168
- 来自 Elea 的陌生人， 109–10
- 主观虚构 146
- 苏尔瓦苏特拉 27–34 , 45 , 50 , 56 , 62 , 99 , 113
- 三段论理论 125
- 符号编号 107
- Taitirīya Samhitā 31
- 塔木德， 21
- 泰勒，A. E. 52
- 泰勒，布鲁克 58
- 技术 64-5
- 时间序列 54
- 特伦斯 192
- 三元组 92–3
- 亚历山大的席恩 30
- 士麦那的席恩 104 , 109 , 110
- 理论 4 , 8
- 论文 26-7
- 蒂博，乔治 29 岁，56 岁
- 托马斯·阿奎那 37 , 138 , 195
- 透特 24
- Thureau-Dangin，第61页
- 时间和空间 15 , 121–2
- Toeplitz 矩阵 122–3 , 215n12
- 超越数 159
- 超限数 118 , 137
- 真相 12 , 36-7 , 38-9
- 图灵, 艾伦 5 , 75 , 131 , 150 , 156 , 169 , 174–5 , 185 , 199–200 , 201n2
- 网络， 198-9
- 韦伯，海因里希 132
- 维尔斯特拉斯, 卡尔 15 , 22 , 81 , 83 , 86 , 96 , 99 , 105 , 106 , 141 , 170
- 威尔，西蒙娜 17 , 27 , 68 , 74 , 130 , 194 , 206n1 , 213n10
- 外尔赫尔曼 88
- Whitehead, A. N. 79 , 83–4 , 86 , 117 , 131 , 138 , 142 , 168–9
- 整数 7 , 22 , 131 , 177–8
- 维纳 、诺伯特54、122、124 _ _
- 维纳-霍普夫方程 122 , 123
- 威尔金森 ，詹姆斯169、172、174、224n12 _ _ _ _
- 世界, 的, 订婚 3
神的数学和人的算法
the mathematics of the gods and the algorithms of men
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